{"id":132,"date":"2025-04-09T17:14:45","date_gmt":"2025-04-09T17:14:45","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/3-1-functions-and-function-notation-college-algebra-2e-openstax\/"},"modified":"2025-08-19T16:32:26","modified_gmt":"2025-08-19T16:32:26","slug":"3-1-functions-and-function-notation","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/3-1-functions-and-function-notation\/","title":{"raw":"3.1 Functions and Function Notation","rendered":"3.1 Functions and Function Notation"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_55f2e8ec-a982-4586-9d48-a2f43d7b4107\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Determine whether a relation represents a function.<\/li>\r\n \t<li>Find the value of a function.<\/li>\r\n \t<li>Determine whether a function is one-to-one.<\/li>\r\n \t<li>Use the vertical line test to identify functions.<\/li>\r\n \t<li>Graph the functions listed in the library of functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137431376\">A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.<\/p>\r\n\r\n<section id=\"fs-id1165133394710\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Determining Whether a Relation Represents a Function<\/h2>\r\n<p id=\"fs-id1165137781542\">A <span id=\"term-00005\" data-type=\"term\">relation<\/span> is a set of ordered pairs. The set of the first components of each <span id=\"term-00006\" class=\"no-emphasis\" data-type=\"term\">ordered pair<\/span> is called the <strong>domain <\/strong>and the set of the second components of each ordered pair is called the <strong>range<\/strong>. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\{(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)\\} [\/latex]<\/p>\r\n<p id=\"fs-id1165133155834\">The domain is [latex] \\{1, 2, 3, 4, 5\\}. [\/latex] The range is [latex] \\{2, 4, 6, 8, 10\\}. [\/latex]<\/p>\r\n<p id=\"fs-id1165134234609\">Note that each value in the domain is also known as an <strong>input<\/strong> value, or <span id=\"term-00007\" data-type=\"term\">independent variable<\/span>, and is often labeled with the lowercase letter [latex] x. [\/latex] Each value in the range is also known as an <strong>output<\/strong> value, or <span id=\"term-00008\" data-type=\"term\">dependent variable<\/span>, and is often labeled lowercase letter [latex] y. [\/latex]<\/p>\r\n<p id=\"fs-id1165137748300\">A function [latex] f [\/latex] is a relation that assigns a single value in the range to each value in the domain<em data-effect=\"italics\">.<\/em> In other words, no <em data-effect=\"italics\">x<\/em>-values are repeated. For our example that relates the first five <span id=\"term-00009\" class=\"no-emphasis\" data-type=\"term\">natural numbers<\/span> to numbers double their values, this relation is a function because each element in the domain, [latex] \\{1, 2, 3, 4, 5\\}, [\/latex] is paired with exactly one element in the range, [latex] \\{2, 4, 6, 8, 10\\}. [\/latex]<\/p>\r\n<p id=\"fs-id1165135421564\">Now let\u2019s consider the set of ordered pairs that relates the terms \u201ceven\u201d and \u201codd\u201d to the first five natural numbers. It would appear as<\/p>\r\n<p style=\"text-align: center;\">[latex] \\{(\\text{odd}, 1), (\\text{even}, 2), (\\text{odd}, 3), (\\text{even}, 4), (\\text{odd}, 5)\\} [\/latex]<\/p>\r\n<p id=\"fs-id1165135419796\">Notice that each element in the domain, [latex] \\{\\text{even}, \\text{odd}\\} [\/latex] is <em data-effect=\"italics\">not<\/em> paired with exactly one element in the range, [latex] \\{1, 2, 3, 4, 5\\}. [\/latex] For example, the term \u201codd\u201d corresponds to three values from the range, [latex] \\{1, 3, 5\\} [\/latex] and the term \u201ceven\u201d corresponds to two values from the range, [latex] \\{2, 4\\} [\/latex] This violates the definition of a function, so this relation is not a function.<\/p>\r\n<p id=\"fs-id1165135176295\">Figure 1 compares relations that are functions and not functions.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_449\" align=\"aligncenter\" width=\"547\"]<img class=\" wp-image-449\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-Fi.-1-300x90.jpg\" alt=\"\" width=\"547\" height=\"164\" \/> Three relations that demonstrate what constitute a function.<br \/>Figure 1 (a) This relationship is a function because each input is associated with a single output. Note that input q and r both give output n. (b) This relationship is also a function. In this case, each input is associated with a single output. (c) This relationship is not a function because input q is associated with two different outputs.[\/caption]\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA <strong>function<\/strong> is a relation in which each possible input value leads to exactly one output value. We say \"the output is a function of the input.\"\r\n\r\nThe <strong>input<\/strong> values make up the\u00a0<strong>domain<\/strong>, and the <strong>output<\/strong> values make up the\u00a0<strong>range<\/strong>.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven a relationship between two quantities, determine whether the relationship is a function.\r\n<ol>\r\n \t<li>Identify the input values.<\/li>\r\n \t<li>Identify the output values.<\/li>\r\n \t<li>If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example I: Determining in Menu Price Lists are Functions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA donut shop in Park Meadows is a local favorite. Their menu, shown below, consists of items and their prices.\r\n\r\n(a) Is price a function of the item?\r\n\r\n(b) Is the item a function of the price?\r\n\r\n<img class=\"wp-image-450 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1-300x144.jpg\" alt=\"\" width=\"479\" height=\"230\" \/>\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a) Let's begin by considering the input as the items on the menu. The output values are then the prices.\r\n\r\n<img class=\"alignnone wp-image-1149 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1-2-300x144.jpg\" alt=\"\" width=\"519\" height=\"249\" \/>\r\n\r\nEach item on the menu has only one price, so the price is a function of the item.\r\n\r\n(b) Two items on the menu have the same price. If we consider the prices to be the input values and the items to be the output, then the same input value could have more than one output associated with it. See the image below.\r\n\r\n<img class=\"wp-image-452 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1.1-300x144.jpg\" alt=\"\" width=\"523\" height=\"251\" \/>\r\n\r\nTherefore, the item is not a function of the price.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 2: Determining if Class Grade Rules are Functions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIn a College Algebra class, the overall percent grade corresponds to a grade point average. Is grade point average a function of the percent grade? Is the percent grade a function of the grade point average? Table 1 shows a possible rule for assigning grade points.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 1<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 11.1111%;\"><strong>Percent grade<\/strong><\/td>\r\n<td style=\"width: 11.1111%;\">0-56<\/td>\r\n<td style=\"width: 11.1111%;\">57-61<\/td>\r\n<td style=\"width: 11.1111%;\">62-66<\/td>\r\n<td style=\"width: 11.1111%;\">67-71<\/td>\r\n<td style=\"width: 11.1111%;\">72-77<\/td>\r\n<td style=\"width: 11.1111%;\">78-86<\/td>\r\n<td style=\"width: 11.1111%;\">87-91<\/td>\r\n<td style=\"width: 11.1111%;\">92-100<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 11.1111%;\"><strong>Grade point average<\/strong><\/td>\r\n<td style=\"width: 11.1111%;\">0.0<\/td>\r\n<td style=\"width: 11.1111%;\">1.0<\/td>\r\n<td style=\"width: 11.1111%;\">1.5<\/td>\r\n<td style=\"width: 11.1111%;\">2.0<\/td>\r\n<td style=\"width: 11.1111%;\">2.5<\/td>\r\n<td style=\"width: 11.1111%;\">3.0<\/td>\r\n<td style=\"width: 11.1111%;\">3.5<\/td>\r\n<td style=\"width: 11.1111%;\">4.0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>For any percent grade earned, there is an associated grade point average, so the grade point average is a function of the percent grade. In other words, if we input the percent grade, the output is a specific grade point average.\r\n\r\nIn the grading system given, there is a range of percent grades that correspond to the same grade point average. For example, students who receive a grade point average of 3.0 could have a variety of percent grades ranging from 78 all the way to 86. Thus, percent grade is not a function of grade point average.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nTable 2 lists five students in a mathematics contest at CCA in order of rank.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Tab;e 2<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\"><strong>Student<\/strong><\/td>\r\n<td style=\"width: 50%;\"><strong>Rank<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">Daniela<\/td>\r\n<td style=\"width: 50%;\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">Amir<\/td>\r\n<td style=\"width: 50%;\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">John<\/td>\r\n<td style=\"width: 50%;\">3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">Mary<\/td>\r\n<td style=\"width: 50%;\">4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">Diego<\/td>\r\n<td style=\"width: 50%;\">5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n(a) Is the rank a function of the student name?\r\n\r\n(b) Is the student name a function of the rank?\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165134474160\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Using Function Notation<\/h3>\r\n<p id=\"fs-id1165133359348\">Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard <span id=\"term-00015\" class=\"no-emphasis\" data-type=\"term\">function notation<\/span> is one representation that facilitates working with functions.<\/p>\r\n<p id=\"fs-id1165137453971\">To represent \u201cheight is a function of age,\u201d we start by identifying the descriptive variables [latex] h [\/latex] for height and [latex] a [\/latex] for age. The letters [latex] f, g [\/latex] and [latex] h [\/latex] are often used to represent functions just as we use [latex] x, y [\/latex] and [latex] z [\/latex] to represent numbers and [latex] A, B [\/latex] and [latex] C [\/latex] to represent sets.<\/p>\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{lllll}h \\; \\text{is} \\; f \\; \\text{of} \\; a &amp;&amp;&amp;&amp;\\text{We name the function } f; \\; \\text{height is a function of age.} \\\\h = f(a) &amp;&amp;&amp;&amp;\\text{We use parentheses to indicate the function input.} \\\\f(a) &amp;&amp;&amp;&amp;\\text{We name the function } f; \\; \\text{the expression is read as ``}f \\; \\text{of } a\\text{.\"} \\\\\\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1165137766965\">Remember, we can use any letter to name the function; the notation [latex] h(a) [\/latex] shows us that [latex] h [\/latex] depends on [latex] a. [\/latex] The value [latex] a [\/latex] must be put into the function [latex] h [\/latex] to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.<\/p>\r\n<p id=\"fs-id1165135436660\">We can also give an algebraic expression as the input to a function. For example [latex] f(a+b) [\/latex] means \u201cfirst add <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em>, and the result is the input for the function <em data-effect=\"italics\">f<\/em>.\u201d The operations must be performed in this order to obtain the correct result.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Function Notation<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe notation\u00a0[latex] y=f(x) [\/latex] defines a function named\u00a0[latex] f. [\/latex] This is read as \"[latex] y [\/latex] is a function of [latex] x. [\/latex]\" The letter\u00a0[latex] x [\/latex] represents the input value, or independent variable. The letter\u00a0[latex] y [\/latex] or\u00a0[latex] f(x) [\/latex] represents the output value, or dependent variable.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3: Using Function Notation for Days in a Month<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse function notation to represent a function whose input is the name of a month and output is the number of days in that month. Assume that the domain does not include leap years.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The number of days in a month is a function of the name of the month, so if we name the function\u00a0[latex] f, [\/latex] we write\u00a0[latex] \\text{days}=f(\\text{month}) [\/latex] or [latex] d=f(m) [\/latex] The name of the month is the input to a \"rule\" that associates a specific number (the output) with each input.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_455\" align=\"aligncenter\" width=\"354\"]<img class=\"wp-image-455 \" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-3.jpg\" alt=\"\" width=\"354\" height=\"205\" \/> Figure 2[\/caption]\r\n\r\nFor example,\u00a0[latex] f(\\text{March})=31 [\/latex] because March has 31 days. The notation\u00a0[latex] d=f(m) [\/latex] reminds us that the number of days, [latex] d [\/latex](the output), is dependent on the name of the month, [latex] m [\/latex](the input).\r\n<h3>Analysis<\/h3>\r\nNote that the inputs of a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 4: Interpreting Function Notation<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA function\u00a0[latex] N=f(y) [\/latex] gives the numbers of police officers, [latex] N [\/latex] in Aurora, Colorado, in year [latex] y. [\/latex] What does\u00a0[latex] f(2005)=300 [\/latex] represent?\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>When we read [latex] f(2005)=300, [\/latex], we see that the input year is 2005. The value for the output, the number of police officers\u00a0[latex] (N), [\/latex] is 300. Remember,\u00a0[latex] N=f(y). [\/latex] The statement\u00a0[latex] f(2005)=300 [\/latex] tells us that in the year 2005 there were 300 police officers in Aurora.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUse function notation to express the weight of a pig in pounds as a function of its age in days [latex] d. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nQ: Instead of notation such as\u00a0[latex] y=f(x), [\/latex] could we use the same symbol for the output as for the function, such as\u00a0[latex] y=y(x), [\/latex] meaning \"[latex] y [\/latex] is a function of [latex] x [\/latex]?\"\r\n\r\n<em>A: Yes, this is often done, especially in applied subjects that use higher math, such as physics and engineering. However, in exploring math itself we like to maintain a distinction between a function such as\u00a0[latex] f [\/latex] which is a rule or procedure, and the output\u00a0[latex] y [\/latex] we get by applying\u00a0[latex] f [\/latex] to a particular input\u00a0[latex] x. [\/latex] This is why we usually use notation such as\u00a0[latex] y=f(x), P=W(d), [\/latex] and so on.<\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137804204\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Representing Functions Using Tables<\/h3>\r\n<p id=\"fs-id1165137648317\">A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values.\u00a0In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship.<\/p>\r\n<p id=\"fs-id1165137761188\">Table 3 lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function\u00a0[latex] f [\/latex] where [latex] D=f(m) [\/latex] identifies months by an integer rather than by name.<\/p>\r\n\r\n<div id=\"Table_01_01_03\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"Table_01_01_03\"><caption>Table 3<\/caption>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\"><strong>Month number, [latex] m [\/latex] <\/strong>(input)<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td data-align=\"center\">3<\/td>\r\n<td data-align=\"center\">4<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td data-align=\"center\">7<\/td>\r\n<td data-align=\"center\">8<\/td>\r\n<td data-align=\"center\">9<\/td>\r\n<td data-align=\"center\">10<\/td>\r\n<td data-align=\"center\">11<\/td>\r\n<td data-align=\"center\">12<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\"><strong>Days in month, [latex] D [\/latex] <\/strong>(output)<\/td>\r\n<td data-align=\"center\">31<\/td>\r\n<td data-align=\"center\">28<\/td>\r\n<td data-align=\"center\">31<\/td>\r\n<td data-align=\"center\">30<\/td>\r\n<td data-align=\"center\">31<\/td>\r\n<td data-align=\"center\">30<\/td>\r\n<td data-align=\"center\">31<\/td>\r\n<td data-align=\"center\">31<\/td>\r\n<td data-align=\"center\">30<\/td>\r\n<td data-align=\"center\">31<\/td>\r\n<td data-align=\"center\">30<\/td>\r\n<td data-align=\"center\">31<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\">\r\n\r\nTable 4 defines a function [latex] Q=g(n). [\/latex] Remember, this notation tells us that [latex] g [\/latex] is the name of the function that takes the input [latex] n [\/latex] and gives the output [latex] Q. [\/latex]\r\n<div id=\"Table_01_01_04\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"Table_01_01_04\"><caption>Table 4<\/caption><colgroup> <col \/> <col data-width=\"25\" \/> <col data-width=\"25\" \/> <col data-width=\"25\" \/> <col data-width=\"25\" \/> <col data-width=\"25\" \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] n [\/latex]<\/td>\r\n<td data-align=\"center\">1<\/td>\r\n<td data-align=\"center\">2<\/td>\r\n<td data-align=\"center\">3<\/td>\r\n<td data-align=\"center\">4<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] Q [\/latex]<\/td>\r\n<td data-align=\"center\">8<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td data-align=\"center\">7<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td data-align=\"center\">8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\">Table 5 displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.<\/div>\r\n<\/div>\r\n<div id=\"Table_01_01_05\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"Table_01_01_05\"><caption>Table 5<\/caption><colgroup> <col \/> <col data-width=\"30\" \/> <col data-width=\"30\" \/> <col data-width=\"30\" \/> <col data-width=\"30\" \/> <col data-width=\"30\" \/> <col data-width=\"30\" \/> <col data-width=\"30\" \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\"><strong>Age in years, <\/strong> [latex] a [\/latex] (input)<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<td data-align=\"center\">6<\/td>\r\n<td data-align=\"center\">7<\/td>\r\n<td data-align=\"center\">8<\/td>\r\n<td data-align=\"center\">9<\/td>\r\n<td data-align=\"center\">10<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\"><strong>Height in inches, <\/strong> [latex] h [\/latex] (output)<\/td>\r\n<td data-align=\"center\">40<\/td>\r\n<td data-align=\"center\">42<\/td>\r\n<td data-align=\"center\">44<\/td>\r\n<td data-align=\"center\">47<\/td>\r\n<td data-align=\"center\">50<\/td>\r\n<td data-align=\"center\">52<\/td>\r\n<td data-align=\"center\">54<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><\/div>\r\n<div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a table of input and output values, determine whether the table represents a function.<\/strong>\r\n<ol>\r\n \t<li>Identify the input and output values.<\/li>\r\n \t<li>Check to see if each input value is paired with only one output value. If so, the table represents a function.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 5: Identifying Tables that Represent Functions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhich table (if any), Table 6, Table 7, or Table 8, represents a function?\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 6<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\"><strong>Input<\/strong><\/td>\r\n<td style=\"width: 50%; text-align: center;\"><strong>Output<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">2<\/td>\r\n<td style=\"width: 50%; text-align: center;\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">5<\/td>\r\n<td style=\"width: 50%; text-align: center;\">3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">8<\/td>\r\n<td style=\"width: 50%; text-align: center;\">6<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 7<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\"><strong>Input<\/strong><\/td>\r\n<td style=\"width: 50%; text-align: center;\"><strong>Output<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">-3<\/td>\r\n<td style=\"width: 50%; text-align: center;\">5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">0<\/td>\r\n<td style=\"width: 50%; text-align: center;\">1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">4<\/td>\r\n<td style=\"width: 50%; text-align: center;\">5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 8<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\"><strong>Input<\/strong><\/td>\r\n<td style=\"width: 50%; text-align: center;\"><strong>Output<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">1<\/td>\r\n<td style=\"width: 50%; text-align: center;\">0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">5<\/td>\r\n<td style=\"width: 50%; text-align: center;\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">5<\/td>\r\n<td style=\"width: 50%; text-align: center;\">4<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Table 6 and Table 7 define functions. In both, each input value corresponds to exactly one output value. Table 8 does not define a function because the input value of 5 corresponds to two different output values.\r\n\r\nWhen a table represents a function, corresponding input and output values can also be specified using function notation.\r\n\r\nThe function represented by Table 6 can be represented by writing\r\n<p style=\"text-align: center;\">[latex] f(2) = 1, f(5) = 3, \\text{and } f(8) = 6 [\/latex]<\/p>\r\nSimilarly, the statements\r\n<p style=\"text-align: center;\">[latex] g(-3) = 5, \\; g(0) = 1, \\text{and } g(4) = 5 [\/latex]<\/p>\r\nrepresent the function in Table 7.\r\n\r\nTable 8 cannot be expressed in a similar way because it does not represent a function.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDoes Table 9 represent a function? Why or why not?\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 9<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\"><strong>Input<\/strong><\/td>\r\n<td style=\"width: 50%; text-align: center;\"><strong>Output<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">1<\/td>\r\n<td style=\"width: 50%; text-align: center;\">10<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">2<\/td>\r\n<td style=\"width: 50%; text-align: center;\">100<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%; text-align: center;\">3<\/td>\r\n<td style=\"width: 50%; text-align: center;\">1000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/section><section id=\"fs-id1165137503241\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Finding Input and Output Values of a Function<\/h2>\r\n<p id=\"fs-id1165137470651\">When we know an input value and want to determine the corresponding output value for a function, we <em data-effect=\"italics\">evaluate<\/em> the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.<\/p>\r\n<p id=\"fs-id1165137735634\">When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function\u2019s formula and <em data-effect=\"italics\">solve<\/em> for the input. Solving can produce more than one solution because different input values can produce the same output value.<\/p>\r\n\r\n<section id=\"fs-id1165137425943\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Evaluation of Functions in Algebraic Forms<\/h3>\r\n<p id=\"fs-id1165137655584\">When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function [latex] f(x)=5-3x^2 [\/latex] can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given the formula for a function, evaluate.<\/strong>\r\n<ol>\r\n \t<li>Substitute the input variable in the formula for the value provided.<\/li>\r\n \t<li>Calculate the result.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 6: Evaluating Functions at Specific Values<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nEvaluate\u00a0[latex] f(x)=x^2+3x-4 [\/latex] at:\r\n\r\n(a) [latex] 2 [\/latex]\r\n\r\n(b) [latex] a [\/latex]\r\n\r\n(c) [latex] a+h [\/latex]\r\n\r\n(d) Now evaluate [latex] \\frac{f(a+h)-f(a)}{h} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Replace the\u00a0[latex] x [\/latex] in the function with each specified value.\r\n\r\n(a) Because the input value is a number 2, we can use simple algebra to simplify.\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{ll} f(2) &amp; =2^2+3(2)-4 \\\\ &amp; = 4+6-4 \\\\ &amp; = 6 \\end{array} [\/latex]<\/p>\r\n(b) In this case, the input value is a letter so we cannot simplify the answer any further.\r\n<p style=\"text-align: center;\">[latex] f(a)=a^2+3a-4 [\/latex]<\/p>\r\n(c) With an input value of [latex] a+h [\/latex], we must use the distributive property.\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{ll} f(a+h) &amp; =(a+h)^2+3(a+h)-4 \\\\ &amp; = a^2+2ah+h^2+3a+3h-4 \\end{array} [\/latex]<\/p>\r\n(d) In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that\r\n<p style=\"text-align: center;\">[latex] f(a+h)=a^2+2ah+h^2+3a+3h-4 [\/latex]<\/p>\r\n<p id=\"fs-id1165135632109\">and we know that<\/p>\r\n<p style=\"text-align: center;\">[latex] f(a)=a^2+3a-4 [\/latex]<\/p>\r\n<p id=\"fs-id1165137767461\">Now we combine the results and simplify.<\/p>\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{lll} \\frac{f(a+h)-f(a)}{h} &amp; =\\frac{(a^2+2ah+h^2+3a+3h-4)+(a^2+3a-4)}{h} \\\\ &amp; = \\frac{2ah+h^2+3h}{h} \\\\ &amp; = \\frac{h(2a+h+3)}{h} &amp; \\text{Factor out} \\ h. \\\\ &amp; = 2a+h+3 &amp; \\text{Simplify.} \\end{array}[\/latex]<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 7: Evaluating Functions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven the function\u00a0[latex] h(p)=p^2+2p, [\/latex] evaluate [latex] h(4). [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>To evaluate [latex] h(4), [\/latex] we substitute the value 4 for the input variable\u00a0[latex] p [\/latex] in the given function.\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{ll} h(p) &amp; =p^2+2p \\\\ h(4) &amp; =(4^2)+2(4) \\\\ &amp; =16+8 \\\\ &amp; =24 \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1165137785006\">Therefore, for an input of 4, we have an output of 24.<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven the function [latex] g(m)=\\sqrt{m-4}, [\/latex] evaluate [latex] g(5). [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 8: Solving Functions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven the function\u00a0[latex] h(p)=p^2+2p, [\/latex] solve for [latex] h(p)=3. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{lll} h(p) &amp; =3 \\\\ p^2+2p &amp; =3 &amp; \\text{Substitute the original function} \\ h(p)=p^2+2p. \\\\ p^2+2p-3 &amp; =0 &amp; \\text{Subtract 3 from each side.} \\\\ (p+3)(p-1) &amp; =0 &amp; \\text{Factor.} \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1165137770370\">If [latex] (p+3)(p-1)=0, [\/latex] either [latex] (p+3)=0 [\/latex] or [latex] (p-1)=0 [\/latex] (or both of them equal 0). We will set each factor equal to 0 and solve for in each case.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{lll} (p+3) &amp; =0, &amp; p=-3 \\\\ (p-1) &amp; =0, &amp; p=1 \\end{array} [\/latex]<\/p>\r\nThis gives us two solutions. The output\u00a0[latex] h(p)=3 [\/latex] when the input is either\u00a0[latex] p=1 [\/latex] or [latex] p=-3 [\/latex] We can also verify by graphing as in Figure 3. The graph verifies that [latex] h(1)=h(-3)=3 [\/latex]\u00a0and [latex] h(4)=24. [\/latex]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_457\" align=\"aligncenter\" width=\"465\"]<img class=\" wp-image-457\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-8-294x300.jpg\" alt=\"\" width=\"465\" height=\"474\" \/> Figure 3[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven the function [latex] g(m)=\\sqrt{m-4}, [\/latex] solve [latex] g(m)=2. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137591827\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Evaluating Functions Expressed in Formulas<\/h3>\r\n<p id=\"fs-id1165137598337\">Some functions are defined by mathematical rules or procedures expressed in <span id=\"term-00016\" class=\"no-emphasis\" data-type=\"term\">equation<\/span> form. If it is possible to express the function output with a <span id=\"term-00017\" class=\"no-emphasis\" data-type=\"term\">formula<\/span> involving the input quantity, then we can define a function in algebraic form. For example, the equation\u00a0[latex] 2n+6p=12 [\/latex] expresses a functional relationship between [latex] n [\/latex] and [latex] p. [\/latex] We can rewrite it to decide if [latex] p [\/latex] is a function of [latex] n. [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a function in equation form, write is algebraic formula.<\/strong>\r\n<ol>\r\n \t<li>Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an expression that involves <em>only<\/em> the input variable.<\/li>\r\n \t<li>Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or from both sides, or multiplying or dividing both sides of the equation by the same quantity.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 9: Finding the Algebraic Form of a Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nExpress the relationship\u00a0[latex] 2n+6p=12 [\/latex] as a function\u00a0[latex] p=f(n), [\/latex] if possible.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>To express the relationship in this form, we need to be able to write the relationship where\u00a0[latex] p [\/latex] is a function of\u00a0[latex] n, [\/latex] which means writing it as [latex] p=[\\text{expression involving}\\hspace{0.25em}n]. [\/latex]\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{lll} 2n+6p &amp; =12 \\\\ 6p &amp; =12-2n &amp; \\text{Subtract} \\ 2n \\ \\text{from both sides.} \\\\ p &amp; =\\frac{12-2n}{6} &amp; \\text{Divide both sides by 6 and simplify.} \\\\ p &amp;=\\frac{12}{6}-\\frac{2n}{6} \\\\ p &amp; = 2-\\frac{1}{3}n \\end{array} [\/latex]<\/p>\r\nTherefore, [latex] p [\/latex] as a function of [latex] n [\/latex] is written as\r\n\r\n[latex] \\hspace{3em}p=f(n)=2-\\frac{1}{3}n [\/latex]\r\n<h3>Analysis<\/h3>\r\nIt is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 10: Expressing the Equation of a Circle as a Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDoes the equation\u00a0[latex] x^2+y^2=1 [\/latex] represent a function with\u00a0[latex] x [\/latex] as input and\u00a0[latex] y [\/latex] as output? If so, express the relationship as a function [latex] y=f(x). [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>First we subtract\u00a0[latex] x^2 [\/latex] from both sides.\r\n<p style=\"text-align: center;\">[latex] y^2=1-x^2 [\/latex]<\/p>\r\nWe now try to solve [latex] y [\/latex] for this equation.\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{ll} y &amp; =\\pm\\sqrt{1-x^2} \\\\ &amp; =+\\sqrt{1-x^2} \\ \\text{and} \\ -\\sqrt{1-x^2} \\end{array}[\/latex]<\/p>\r\nWe get two outputs corresponding to the same input, so this relationship cannot be represented as a single function [latex] y=f(x). [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"body\">\r\n<div id=\"fs-id1165135378843\" class=\"unnumbered\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165135378845\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIf\u00a0[latex] x=8y^3=0, [\/latex] express\u00a0[latex] y [\/latex] as a function of [latex] x. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Q: Are there relationships expressed by an equation that do represent a function but which still cannot be represented by an algebraic formula?<\/strong>\r\n\r\n<em>A: Yes, this can happen. For example, given the equation\u00a0[latex] x=y+2^y, [\/latex] if we want to express [latex] y [\/latex] as a function of [latex] x, [\/latex] there is no simple algebraic formula involving only\u00a0[latex] x [\/latex] that equals\u00a0[latex] y. [\/latex] However, each\u00a0[latex] x [\/latex] does determine a unique value for\u00a0[latex] y, [\/latex] and there are mathematical procedures by which [latex] y [\/latex] can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for\u00a0[latex] y [\/latex] as a function of\u00a0[latex] x, [\/latex] even though the formula cannot be written explicitly.<\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/section><\/div>\r\n<section id=\"fs-id1165137648450\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Evaluating a Function Given in Tabular Form<\/h3>\r\n<p id=\"fs-id1165135186424\">As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy\u2019s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.<\/p>\r\n<p id=\"fs-id1165135186427\" class=\"has-noteref\">The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See Table 10.[footnote]http:\/\/www.kgbanswers.com\/how-long-is-a-dogs-memory-span\/4221590. Accessed 3\/24\/2014.[\/footnote]<sup id=\"footnote-ref2\" data-type=\"footnote-number\"><\/sup><\/p>\r\n\r\n<div id=\"Table_01_01_10\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"Table_01_01_10\"><caption>Table 10<\/caption><colgroup> <col data-width=\"85\" data-align=\"center\" \/> <col data-align=\"center\" \/><\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\" scope=\"col\" data-align=\"center\">Pet<\/th>\r\n<th style=\"text-align: center;\" scope=\"col\" data-align=\"center\">Memory span in hours<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\" data-align=\"center\">Puppy<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\">0.008<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\" data-align=\"center\">Adult dog<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\">0.083<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\" data-align=\"center\">Cat<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\">16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\" data-align=\"center\">Goldfish<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\">2160<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\" data-align=\"center\">Beta fish<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\">3600<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\">At times, evaluating a function in table form may be more useful than using equations. Here let us call the function\u00a0[latex] P. [\/latex] The domain of the function is the type of pet and the range is a real number representing the number of hours the pet\u2019s memory span lasts. We can evaluate the function [latex] P [\/latex] at the input value of \u201cgoldfish.\u201d We would write [latex] P(\\text{goldfish})=2160. [\/latex] Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function [latex] P [\/latex] seems ideally suited to this function, more so than writing it in paragraph or function form.<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a function represented by a table, identify specific output and input values.<\/strong>\r\n<ol>\r\n \t<li>Find the given input in the row (or column) of input values.<\/li>\r\n \t<li>Identify the corresponding output value paired with that input value.<\/li>\r\n \t<li>Find the given output values in the row (or column) of output values, noting every time that output value appears.<\/li>\r\n \t<li>Identify the input value(s) corresponding to the given output value.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 11: Evaluating and Solving a Tabular Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUsing Table 11,\r\n\r\n(a) Evaluate [latex] g(3). [\/latex]\r\n\r\n(b) Solve [latex] g(n)-6. [\/latex]\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 11<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 16.6667%;\">[latex] n [\/latex]<\/td>\r\n<td style=\"width: 16.6667%;\">1<\/td>\r\n<td style=\"width: 16.6667%;\">2<\/td>\r\n<td style=\"width: 16.6667%;\">3<\/td>\r\n<td style=\"width: 16.6667%;\">4<\/td>\r\n<td style=\"width: 16.6667%;\">5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.6667%;\">[latex] g(n) [\/latex]<\/td>\r\n<td style=\"width: 16.6667%;\">8<\/td>\r\n<td style=\"width: 16.6667%;\">6<\/td>\r\n<td style=\"width: 16.6667%;\">7<\/td>\r\n<td style=\"width: 16.6667%;\">6<\/td>\r\n<td style=\"width: 16.6667%;\">8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a) Evaluating\u00a0[latex] g(3) [\/latex] means determining the output value of the function\u00a0[latex] g [\/latex] for the input value of\u00a0[latex] n=3. [\/latex] The table output value corresponding to\u00a0[latex] n=3 [\/latex] is 7, so [latex] g(3)=7. [\/latex]\r\n\r\n(b) Solving\u00a0[latex] g(n)=6 [\/latex] means identifying the input values,\u00a0[latex] n, [\/latex] that produce an output of 6. The table shows two solutions:\u00a0[latex] 2 [\/latex] and [latex] 4. [\/latex]\r\n<table class=\"grid\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 171px;\">[latex] n [\/latex]<\/td>\r\n<td style=\"width: 97px;\">1<\/td>\r\n<td style=\"width: 98px;\">2<\/td>\r\n<td style=\"width: 97px;\">3<\/td>\r\n<td style=\"width: 98px;\">4<\/td>\r\n<td style=\"width: 97px;\">5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 171px;\">[latex] g(n) [\/latex]<\/td>\r\n<td style=\"width: 97px;\">8<\/td>\r\n<td style=\"width: 98px;\">6<\/td>\r\n<td style=\"width: 97px;\">7<\/td>\r\n<td style=\"width: 98px;\">6<\/td>\r\n<td style=\"width: 97px;\">8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhen we input 2 into the function\u00a0[latex] g, [\/latex] our output is 6. When we input 4 into the function\u00a0[latex] g, [\/latex] our output is also 6.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUsing the table from <strong>Example 11: Evaluating and Solving a Tabular Function<\/strong> above, evaluate [latex] g(1). [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165135696152\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Finding Function Values from a Graph<\/h3>\r\n<p id=\"fs-id1165137779152\">Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 12: Reading Function Values from a Graph<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven the graph in Figure 4,\r\n\r\n(a) Evaluate [latex] f(2). [\/latex]\r\n\r\n(b) Solve [latex] f(x)=4. [\/latex]\r\n\r\n[caption id=\"attachment_459\" align=\"aligncenter\" width=\"505\"]<img class=\" wp-image-459\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-4-300x283.jpg\" alt=\"\" width=\"505\" height=\"476\" \/> Figure 4[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a) To evaluate\u00a0[latex] f(2), [\/latex] locate the point on the curve where\u00a0[latex] x=2, [\/latex] then read the y-coordinate of that point. The point has coordinates\u00a0[latex] (2, 1), [\/latex] so\u00a0[latex] f(2)=1. [\/latex] See Figure 5.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_460\" align=\"aligncenter\" width=\"505\"]<img class=\" wp-image-460\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-5-300x282.jpg\" alt=\"\" width=\"505\" height=\"475\" \/> Figure 5[\/caption]\r\n\r\n(b) To solve\u00a0[latex] f(x)=4, [\/latex] we find the output value [latex] 4 [\/latex] on the vertical axis. Moving horizontally along the line\u00a0[latex] y=4, [\/latex] we locate two points of the curve with output value [latex] 4: (-1, 4) [\/latex] and\u00a0[latex] (3, 4). [\/latex] These points represent the two solutions to\u00a0[latex] f(x)=4: -1 [\/latex] or\u00a0[latex] 3. [\/latex] This means\u00a0[latex] f(-1)=4 [\/latex] and [latex]- f(3)=4 [\/latex] or when the input is\u00a0[latex] -1 [\/latex] or\u00a0[latex] 3 [\/latex] the output is\u00a0[latex] 4. [\/latex] See Figure 6.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_461\" align=\"aligncenter\" width=\"491\"]<img class=\" wp-image-461\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-6-296x300.jpg\" alt=\"\" width=\"491\" height=\"498\" \/> Figure 6[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUsing Figure 4 above, solve [latex] f(x)=1. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135422920\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Determining Whether a Function is One-to-One<\/h2>\r\n<div>\r\n<div>\r\n\r\nSome functions have a given output value that corresponds to two or more input values. For example, in the case chart shown in the figure at the beginning of this chapter, the world-wide cases were 7,000,000 for three different weeks, meaning that there were three different input values that all resulted in the same output value of 7,000,000.\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135245630\">However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in <a class=\"autogenerated-content\" href=\"3-1-functions-and-function-notation#Table_01_01_13\">Table 12<\/a>.<\/p>\r\n\r\n<div id=\"Table_01_01_13\" class=\"os-table\">\r\n<table class=\"grid\" style=\"height: 70px;\" data-id=\"Table_01_01_13\"><caption>Table 12<\/caption><colgroup> <col data-align=\"center\" \/> <col data-align=\"center\" \/><\/colgroup>\r\n<thead>\r\n<tr style=\"height: 15px;\">\r\n<th style=\"height: 15px; width: 258.083px; text-align: center;\" scope=\"col\" data-align=\"center\">Letter grade<\/th>\r\n<th style=\"height: 15px; width: 402.483px; text-align: center;\" scope=\"col\" data-align=\"center\">Grade point average<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 258.083px; text-align: center;\" data-align=\"center\">A<\/td>\r\n<td style=\"height: 15px; width: 402.483px; text-align: center;\" data-align=\"center\">4.0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 258.083px; text-align: center;\" data-align=\"center\">B<\/td>\r\n<td style=\"height: 15px; width: 402.483px; text-align: center;\" data-align=\"center\">3.0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; width: 258.083px; text-align: center;\" data-align=\"center\">C<\/td>\r\n<td style=\"height: 15px; width: 402.483px; text-align: center;\" data-align=\"center\">2.0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 10px;\">\r\n<td style=\"height: 10px; width: 258.083px; text-align: center;\" data-align=\"center\">D<\/td>\r\n<td style=\"height: 10px; width: 402.483px; text-align: center;\" data-align=\"center\">1.0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\">This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165137628999\">To visualize this concept, let\u2019s look again at the two simple functions sketched in Figure 1<strong>(a) <\/strong>and Figure 1<strong>(b)<\/strong>. The function in part (a) shows a relationship that is not a one-to-one function because inputs\u00a0[latex] q [\/latex] and [latex] r [\/latex] both give output [latex] n. [\/latex] The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">One-to-One Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA <strong>one-to-one function<\/strong> is a function in which each output value corresponds to exactly one input value.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 13: Determining Whether a Relationship is a One-to-One Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIs the area of a circle a function of its radius? If yes, is the function one-to-one?\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>A circle of radius\u00a0[latex] r [\/latex] has a unique area measure given by\u00a0[latex] A=\\pi r^2 [\/latex] so for any input [latex] r, [\/latex] there is only one output,\u00a0[latex] A. [\/latex] The area is a function of radius [latex] r. [\/latex]\r\n\r\nIf the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure\u00a0[latex] A [\/latex] is given by the formula [latex] A=\\pi r^2. [\/latex]\u00a0Because areas and radii are positive numbers, there is exactly one solution:\u00a0[latex] \\sqrt{\\frac{A}{\\pi}}. [\/latex] So the area of a circle is a one-to-one function of the circle's radius.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #9<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n(a) Is a balance a function of a bank account number?\r\n\r\n(b) Is a bank account number a function of the balance?\r\n\r\n(c) Is a balance a one-to-one function of the bank account number?\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #10<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nEvaluate the following:\r\n\r\n(a) If each percent grade earned in a course translates to one letter grade, is the letter grade a function of the percent grade?\r\n\r\n(b) If so, is the function one-to-one?\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135435781\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using the Vertical Line Test<\/h2>\r\n<p id=\"fs-id1165135435786\">As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.<\/p>\r\n<p id=\"fs-id1165137637786\">The most common graphs name the input value\u00a0[latex] x [\/latex] and the output value [latex] y, [\/latex] and we say [latex] y [\/latex] is a function of [latex] x, [\/latex] or [latex] y=f(x) [\/latex] when the function is named [latex] f. [\/latex] The graph of the function is the set of all points [latex] (x, y) [\/latex] in the plane that satisfies the equation [latex] y=f(x). [\/latex] If the function is defined for only a few input values, then the graph of the function is only a few points, where the <em data-effect=\"italics\">x<\/em>-coordinate of each point is an input value and the <em data-effect=\"italics\">y<\/em>-coordinate of each point is the corresponding output value. For example, the black dots on the graph in Figure 7 tell us that [latex] f(0)=2 [\/latex] and [latex] f(6)=1. [\/latex] However, the set of all points [latex] (x, y) [\/latex] satisfying [latex] y=f(x) [\/latex] is a curve. The curve shown includes [latex] (0, 2) [\/latex] and [latex] (6, 1) [\/latex] because the curve passes through those points.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_462\" align=\"aligncenter\" width=\"604\"]<img class=\" wp-image-462\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-7-300x220.jpg\" alt=\"\" width=\"604\" height=\"443\" \/> Figure 7[\/caption]\r\n<p id=\"fs-id1165137737620\">The <strong><span id=\"term-00020\" data-type=\"term\">vertical line test<\/span><\/strong> can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does <em data-effect=\"italics\">not<\/em> define a function because a function has only one output value for each input value. See Figure 8.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_463\" align=\"aligncenter\" width=\"708\"]<img class=\" wp-image-463\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-8-300x103.jpg\" alt=\"\" width=\"708\" height=\"243\" \/> Figure 8[\/caption]\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a graph, use the vertical line test to determine if the graph represents a function.<\/strong>\r\n<ol>\r\n \t<li>Inspect the graph to see if any vertical line drawn would intersect the curve more than once.<\/li>\r\n \t<li>Is there is any such line, determine that the graph does not represent a function.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 14: Applying the Vertical Line Test<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhich of the graphs in Figure 9 represent(s) a function [latex] y=f(x)? [\/latex]\r\n\r\n[caption id=\"attachment_464\" align=\"aligncenter\" width=\"561\"]<img class=\" wp-image-464\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-9-300x122.jpg\" alt=\"\" width=\"561\" height=\"228\" \/> Figure 9[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>If any vertical line intersects the graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of Figure 9. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, as most x-values, a vertical line would intersect the graph at more than one point, as shown in Figure 10.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_465\" align=\"aligncenter\" width=\"549\"]<img class=\" wp-image-465\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-10-300x274.jpg\" alt=\"\" width=\"549\" height=\"501\" \/> Figure 10[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #11<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDoes the graph in Figure 11 represent a function?\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_466\" align=\"aligncenter\" width=\"515\"]<img class=\" wp-image-466\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-11-287x300.jpg\" alt=\"\" width=\"515\" height=\"538\" \/> Figure 11[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137610952\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using the Horizontal Line Test<\/h2>\r\n<p id=\"fs-id1165137871503\">Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the <strong><span id=\"term-00021\" data-type=\"term\">horizontal line test<\/span><\/strong>. Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.<\/strong>\r\n<ul>\r\n \t<li>Inspect the graph to see if a horizontal line drawn would intersect the curve more than once,<\/li>\r\n \t<li>Is there is any such line, determine that the function is not one-to-one.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 15: Applying the Horizontal Line Test<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nConsider the functions shown in Figure 9(a) and Figure 9(b). Are either of the functions one-to-one?\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The function in Figure 9(a) is not one-to-one. The horizontal line shown in Figure 12 intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points).\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_468\" align=\"aligncenter\" width=\"594\"]<img class=\" wp-image-468\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-12-290x300.jpg\" alt=\"\" width=\"594\" height=\"614\" \/> Figure 12[\/caption]\r\n\r\nThe function in Figure 9(b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #12<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIs the graph shown in Figure 9(c) one-to-one?\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135545919\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Identifying Basic Toolkit Functions<\/h2>\r\n<p id=\"fs-id1165137698132\">In this text, we will be exploring functions\u2014the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our \u201ctoolkit functions,\u201d which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use\u00a0[latex] x [\/latex] as the input variable and [latex] y=f(x) [\/latex] as the output variable.<\/p>\r\n<p id=\"fs-id1165135591070\">We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown in Table 13.<\/p>\r\n\r\n<div id=\"Table_01_01_14\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"Table_01_01_14\"><caption>Table 13<\/caption><colgroup> <col data-align=\"center\" \/> <col data-align=\"center\" \/> <col data-align=\"center\" \/><\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\" colspan=\"3\" scope=\"colgroup\">Toolkit Functions<\/th>\r\n<\/tr>\r\n<tr>\r\n<th style=\"text-align: center;\" scope=\"col\" data-align=\"center\">Name<\/th>\r\n<th style=\"text-align: center;\" scope=\"col\" data-align=\"center\">Function<\/th>\r\n<th style=\"text-align: center;\" scope=\"col\" data-align=\"center\">Graph<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\" data-align=\"center\">Constant<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\">[latex] f(x)-c, [\/latex] where [latex] c [\/latex] is a constant<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\"><span id=\"fs-id1165137643159\" data-type=\"media\" data-alt=\"Graph of a constant function.\"><img class=\"alignnone size-medium wp-image-469\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-constant-300x179.jpg\" alt=\"\" width=\"300\" height=\"179\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\" data-align=\"center\">Identity<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\">[latex] f(x)=x [\/latex]<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\"><span id=\"fs-id1165137811013\" data-type=\"media\" data-alt=\"Graph of a straight line.\">\r\n<img class=\"alignnone size-medium wp-image-470\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-identity-300x179.jpg\" alt=\"\" width=\"300\" height=\"179\" \/>\r\n<\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\" data-align=\"center\">Absolute value<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\">[latex] f(x)=|x| [\/latex]<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\"><span id=\"fs-id1165135195221\" data-type=\"media\" data-alt=\"Graph of absolute function.\">\r\n<img class=\"alignnone size-medium wp-image-471\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-absolute-300x179.jpg\" alt=\"\" width=\"300\" height=\"179\" \/>\r\n<\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\" data-align=\"center\">Quadratic<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\">[latex] f(x)=x^2 [\/latex]<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\"><span id=\"fs-id1165137501903\" data-type=\"media\" data-alt=\"Graph of a parabola.\">\r\n<img class=\"alignnone size-medium wp-image-472\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-quadratic-300x179.jpg\" alt=\"\" width=\"300\" height=\"179\" \/>\r\n<\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\" data-align=\"center\">Cubic<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\">[latex] f(x)=x^3 [\/latex]<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\"><span id=\"fs-id1165137722123\" data-type=\"media\" data-alt=\"Graph of f(x) = x^3.\">\r\n<img class=\"alignnone size-medium wp-image-473\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-cubic-300x179.jpg\" alt=\"\" width=\"300\" height=\"179\" \/>\r\n<\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\" data-align=\"center\">Reciprocal<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\">[latex] f(x)=\\frac{1}{x} [\/latex]<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\"><span id=\"fs-id1165134544980\" data-type=\"media\" data-alt=\"Graph of f(x)=1\/x.\">\r\n<img class=\"alignnone size-medium wp-image-474\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-reciprocal-300x179.jpg\" alt=\"\" width=\"300\" height=\"179\" \/>\r\n<\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\" data-align=\"center\">Reciprocal squared<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\">[latex] f(x)=\\frac{1}{x^2} [\/latex]<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\"><span id=\"fs-id1165137647610\" data-type=\"media\" data-alt=\"Graph of f(x)=1\/x^2.\">\r\n<img class=\"alignnone size-medium wp-image-475\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-reciprocal-square-300x179.jpg\" alt=\"\" width=\"300\" height=\"179\" \/>\r\n<\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\" data-align=\"center\">Square root<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\">[latex] f(x)=\\sqrt{x} [\/latex]<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\"><span id=\"fs-id1165137863670\" data-type=\"media\" data-alt=\"Graph of f(x)=sqrt(x).\">\r\n<img class=\"alignnone size-medium wp-image-476\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-square-300x179.jpg\" alt=\"\" width=\"300\" height=\"179\" \/>\r\n<\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"text-align: center;\" data-align=\"center\">Cube root<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\">[latex] f(x)=\\sqrt[3]{x} [\/latex]<\/td>\r\n<td style=\"text-align: center;\" data-align=\"center\"><span id=\"fs-id1165137838612\" data-type=\"media\" data-alt=\"Graph of f(x)=x^(1\/3).\">\r\n<img class=\"alignnone size-medium wp-image-477\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-cube-300x179.jpg\" alt=\"\" width=\"300\" height=\"179\" \/>\r\n<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><\/div>\r\n<div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Media<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAccess the following online resources for additional instruction and practice with functions.\r\n<ul>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=zT69oxcMhPw\/\">Determine if a Relation is a Function<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=gO5WN9g1fJo\">Vertical Line Test<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=sW9-zBeQpCU\">Introduction to Functions<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=5Z8DaZPJLKY\/\">Vertical Line Test on Graph<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=QFOJmevha_Y&amp;feature=youtu.be\/\">One-to-one Functions<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=tbSGdcSN8RE&amp;feature=youtu.be\/\">Graphs as One-to-one Functions<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\r\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">3.1 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1165137737761\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1165137432988\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1165137432993\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137432995\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137432993-solution\">1<\/a><span class=\"os-divider\">. <\/span>What is the difference between a relation and a function?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137870912\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137870914\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>What is the difference between the input and the output of a function?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137870922\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134118508\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137870922-solution\">3<\/a><span class=\"os-divider\">. <\/span>Why does the vertical line test tell us whether the graph of a relation represents a function?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135570273\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135570275\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>How can you determine if a relation is a one-to-one function?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134391600\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134391602\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134391600-solution\">5<\/a><span class=\"os-divider\">. <\/span>Why does the horizontal line test tell us whether the graph of a function is one-to-one?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165134080937\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id1165134080942\">For the following exercises, determine whether the relation represents a function.<\/p>\r\n\r\n<div id=\"fs-id1165134080945\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134080947\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex] \\{(a, b), (c, d), (a, c)\\} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135570225\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135570227\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135570225-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex] \\{(a, b), (b, c), (c, c)\\} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135318984\">For the following exercises, determine whether the relation represents [latex] y [\/latex] as a function of [latex] x. [\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165137841682\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137841684\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex] 5x+2y=10 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137925514\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135628498\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137925514-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex] y=x^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137942483\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137722399\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex] x=y^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135394228\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135394230\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135394228-solution\">11<\/a><span class=\"os-divider\">.<\/span> [latex] 3x^2+y=14 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135675195\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135173422\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex] 2x+y^2=6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137864148\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134085981\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137864148-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex] y=-2x^2+40x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137679356\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137924374\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex] y=\\frac{1}{x} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137661060\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137661062\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137661060-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex] x=\\frac{3y+5}{7y-1} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135581214\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135581216\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex] x=\\sqrt{1-y^2} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133202429\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133202431\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165133202429-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex] y=\\frac{3x+5}{7x-1} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134042772\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134042774\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex] x^2+y^2=9 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137887434\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137887436\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137887434-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex] 2xy=1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137658561\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137658563\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex] x=y^3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137777686\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137777688\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137777686-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex] y=x^3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137892513\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137892515\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex] y=\\sqrt{1-x^2} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134216874\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134216877\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134216874-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex] x=\\pm\\sqrt{1-y} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135538770\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135538772\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex] y=\\pm\\sqrt{1-x} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135203596\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135203598\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135203596-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex] y^2=x^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137599984\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137599986\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex] y^3=x^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165134066606\">For the following exercises, evaluate [latex] f(-3, f(2), f(-a), -f(a), f(a+h) [\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165137431335\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137431337\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137431335-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=2x-5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137727203\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137603598\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=-5x^2+2x-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137844088\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137844090\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137844088-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\sqrt{2-x}+5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135697840\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135697842\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{6x-1}{5x+2} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135453854\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135453856\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135453854-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=|x-1|+|x+1| [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135195666\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135195668\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span>Given the function [latex] g(x)=5-x^2, [\/latex] evaluate [latex] \\frac{g(x+h)-g(x)}{h}, h\\not=0. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135579705\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135579707\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135579705-solution\">33<\/a><span class=\"os-divider\">. <\/span>Given the function [latex] g(x)=x^2+2x, [\/latex] evaluate [latex] \\frac{g(x)-g(a)}{x-a}, x\\not=a. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134036847\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134036849\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span>Given the function [latex] k(t)=2t-1: [\/latex]\r\n\r\n(a) Evaluate [latex] k(2). [\/latex]\r\n<div class=\"os-problem-container\">\r\n\r\n(b) Solve [latex] k(t)=7. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134155170\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134155172\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134155170-solution\">35<\/a><span class=\"os-divider\">. <\/span>Given the function [latex] f(x)=8-3x: [\/latex]\r\n\r\n(a) Evaluate [latex] f(-2). [\/latex]\r\n<div class=\"os-problem-container\">\r\n\r\n(b) Solve [latex] f(x)=-1. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137935719\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135388490\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span>Given the function [latex]- p(c)=c^2+c: [\/latex]\r\n\r\n(a) Evaluate [latex] p(-3). [\/latex]\r\n<div class=\"os-problem-container\">\r\n\r\n(b) Solve [latex] p(c)=2. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135361357\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135361359\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135361357-solution\">37<\/a><span class=\"os-divider\">. <\/span>Given the function [latex] f(x)=x^2-3x: [\/latex]\r\n\r\n(a) Evaluate [latex] f(5). [\/latex]\r\n<div class=\"os-problem-container\">\r\n\r\n(b) Solve [latex] f(x)=4. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137833947\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137833949\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span>Given the function [latex] f(x)=\\sqrt{x+2}: [\/latex]\r\n\r\n(a) Evaluate [latex] f(7). [\/latex]\r\n<div class=\"os-problem-container\">\r\n\r\n(b) Solve [latex] f(x)=4. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137433542\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137433544\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137433542-solution\">39<\/a><span class=\"os-divider\">. <\/span>Consider the relationship [latex] 3r+2t=18. [\/latex]\r\n<div class=\"os-problem-container\">\r\n\r\n(a) Write the relationship as a function [latex] r=f(t). [\/latex]\r\n\r\n(b) Evaluate [latex] f(-3). [\/latex]\r\n\r\n(c) Solve [latex] f(t)=2. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165135664071\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphical<\/h3>\r\n<p id=\"fs-id1165135664077\">For the following exercises, use the vertical line test to determine which graphs show relations that are functions.<\/p>\r\n\r\n<div id=\"fs-id1165135455987\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135455989\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<img class=\"alignnone size-medium wp-image-1153\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.40-288x300.webp\" alt=\"\" width=\"288\" height=\"300\" \/>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137527641\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137847086\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137527641-solution\">41<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165137847091\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1154\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.41-288x300.webp\" alt=\"\" width=\"288\" height=\"300\" \/><\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135332512\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133336399\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165133336405\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1155\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.42-288x300.webp\" alt=\"\" width=\"288\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137742393\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137742395\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137742393-solution\">43<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165137597394\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1156\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.43-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135386379\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135386381\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165135386387\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1157\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.44-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137749974\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137439464\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137749974-solution\">45<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165137439470\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1158\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.45-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137399704\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137399706\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165135704896\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1159\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.46-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137883764\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137883767\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137883764-solution\">47<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165137883773\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1160\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.47-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134497159\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134497161\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165134497168\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1161\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.48-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135496435\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135496437\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135496435-solution\">49<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165134234204\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1162\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.49-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137911653\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137911656\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165137786191\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1163\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.50-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135593325\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135593327\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135593325-solution\">51<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165135593333\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"> <img class=\"alignnone size-medium wp-image-1165\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.51-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134240968\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134054028\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1165134054030\">Given the following graph,<\/p>\r\n(a) Evaluate [latex] f(-1). [\/latex]\r\n\r\n(b) Solve for [latex] f(x)=3. [\/latex]\r\n\r\n<span id=\"fs-id1165137834413\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1166\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.52-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135632092\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135632095\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135632092-solution\">53<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1165137861992\">Given the following graph,<\/p>\r\n(a) Evaluate [latex] f(0). [\/latex]\r\n\r\n(b) Solve for [latex] f(x)=-3. [\/latex]\r\n\r\n<span id=\"fs-id1165135567425\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1167\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.53-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134325868\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134325870\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1165134325872\">Given the following graph,<\/p>\r\n(a) Evaluate [latex] f(4). [\/latex]\r\n\r\n(b) Solve for [latex] f(x)=1. [\/latex]\r\n\r\n<span id=\"fs-id1165135575950\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1168\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.54-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135531627\">For the following exercises, determine if the given graph is a one-to-one function.<\/p>\r\n\r\n<div id=\"fs-id1165135541711\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135541713\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135541711-solution\">55<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165135541720\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a circle.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1169\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.55-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133085674\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134380351\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165134380356\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a parabola.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1170\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.56-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134037560\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134037562\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134037560-solution\">57<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165134037568\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a rotated cubic function.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1171\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.57-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134031248\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134031250\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165134031257\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of half of 1\/x.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1172\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.58-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134394579\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134394581\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134394579-solution\">59<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165135457089\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a one-to-one function.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1173\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.59-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165135342204\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Numeric<\/h3>\r\n<p id=\"fs-id1165133324912\">For the following exercises, determine whether the relation represents a function.<\/p>\r\n\r\n<div id=\"fs-id1165133324915\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133324917\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span> [latex] \\{(-1, -1), (-2, -2), (-3, -3)\\} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135245507\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135245509\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135245507-solution\">61<\/a><span class=\"os-divider\">. <\/span> [latex] \\{(3, 4), (4, 5), (5, 6)\\} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135381342\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137724837\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span> [latex] \\{(2, 5), (7, 11), (15, 8), (7, 9)\\} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165133260452\">For the following exercises, determine if the relation represented in table form represents [latex] y [\/latex] as a function of [latex] x. [\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165137644802\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137644804\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137644802-solution\">63<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n<div id=\"fs-id1165137644806\" class=\"os-table first-element\">\r\n<table class=\"grid\" data-id=\"fs-id1165137644806\" data-label=\"\"><colgroup> <col \/> <col data-width=\"50\" \/> <col data-width=\"50\" \/> <col data-width=\"50\" \/> <\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<td data-align=\"center\">10<\/td>\r\n<td data-align=\"center\">15<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] y [\/latex]<\/td>\r\n<td data-align=\"center\">3<\/td>\r\n<td data-align=\"center\">8<\/td>\r\n<td data-align=\"center\">14<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137771740\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137771742\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">64<\/span><span class=\"os-divider\">.<\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n<div id=\"fs-id1165137771744\" class=\"os-table first-element\">\r\n<table class=\"grid\" data-id=\"fs-id1165137771744\" data-label=\"\"><colgroup> <col \/> <col data-width=\"50\" \/> <col data-width=\"50\" \/> <col data-width=\"50\" \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td data-align=\"center\">5<\/td>\r\n<td data-align=\"center\">10<\/td>\r\n<td data-align=\"center\">15<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] y [\/latex]<\/td>\r\n<td data-align=\"center\">3<\/td>\r\n<td data-align=\"center\">8<\/td>\r\n<td data-align=\"center\">8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137758640\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137758643\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137758640-solution\">65<\/a><span class=\"os-divider\">.<\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n<div id=\"fs-id1165137758645\" class=\"os-table first-element\">\r\n<table class=\"grid\" style=\"height: 156px;\" data-id=\"fs-id1165137758645\" data-label=\"\"><colgroup> <col \/> <col data-width=\"50\" \/> <col data-width=\"50\" \/> <col data-width=\"50\" \/><\/colgroup>\r\n<tbody>\r\n<tr style=\"height: 78px;\">\r\n<td style=\"height: 78px; width: 138.783px;\" data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td style=\"height: 78px; width: 135.517px;\" data-align=\"center\">5<\/td>\r\n<td style=\"height: 78px; width: 179.15px;\" data-align=\"center\">10<\/td>\r\n<td style=\"height: 78px; width: 182.183px;\" data-align=\"center\">10<\/td>\r\n<\/tr>\r\n<tr style=\"height: 78px;\">\r\n<td style=\"height: 78px; width: 138.783px;\" data-align=\"center\">[latex] y [\/latex]<\/td>\r\n<td style=\"height: 78px; width: 135.517px;\" data-align=\"center\">3<\/td>\r\n<td style=\"height: 78px; width: 179.15px;\" data-align=\"center\">8<\/td>\r\n<td style=\"height: 78px; width: 182.183px;\" data-align=\"center\">14<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135641701\">For the following exercises, use the function [latex] f [\/latex] represented in the table below.<\/p>\r\n\r\n<div id=\"eip-reftable\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"eip-reftable\"><caption>Table 14<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 92px;\">[latex] x [\/latex]<\/td>\r\n<td style=\"width: 61px;\">0<\/td>\r\n<td style=\"width: 64px;\">1<\/td>\r\n<td style=\"width: 46px;\">2<\/td>\r\n<td style=\"width: 64px;\">3<\/td>\r\n<td style=\"width: 65px;\">4<\/td>\r\n<td style=\"width: 47px;\">5<\/td>\r\n<td style=\"width: 65px;\">6<\/td>\r\n<td style=\"width: 64px;\">7<\/td>\r\n<td style=\"width: 58px;\">8<\/td>\r\n<td style=\"width: 62px;\">9<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 92px;\">[latex] f(x) [\/latex]<\/td>\r\n<td style=\"width: 61px;\">74<\/td>\r\n<td style=\"width: 64px;\">28<\/td>\r\n<td style=\"width: 46px;\">1<\/td>\r\n<td style=\"width: 64px;\">53<\/td>\r\n<td style=\"width: 65px;\">56<\/td>\r\n<td style=\"width: 47px;\">3<\/td>\r\n<td style=\"width: 65px;\">36<\/td>\r\n<td style=\"width: 64px;\">45<\/td>\r\n<td style=\"width: 58px;\">14<\/td>\r\n<td style=\"width: 62px;\">47<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>Evaluate [latex] f(3). [\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137453742\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137453744\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137453742-solution\">67<\/a><span class=\"os-divider\">. <\/span>Solve [latex] f(x)=1. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137757773\">For the following exercises, evaluate the function [latex] f [\/latex] at the values [latex] f(-2), f(-1), f(0), f(1) [\/latex] and [latex] f(2). [\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165135581074\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135581076\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=4-2x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137812524\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137812526\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137812524-solution\">69<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=8-3x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135445749\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135445751\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">70<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=8x^2-7x+3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137937596\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135181211\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137937596-solution\">71<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=3+\\sqrt{x+3} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134573828\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134573830\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">72<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{x-2}{x+3} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133248574\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133248576\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165133248574-solution\">73<\/a><span class=\"os-divider\">. <\/span> [latex] f(x0=3^x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135306461\">For the following exercises, evaluate the expressions, given functions [latex] f, g [\/latex] and [latex] h: [\/latex]<\/p>\r\n[latex] f(x)=3x-2 [\/latex]\r\n[latex] g(x)=5-x^2 [\/latex]\r\n[latex] h(x)=-2x^2+3x-1 [\/latex]\r\n<div id=\"fs-id1165135575197\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135575199\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">74<\/span><span class=\"os-divider\">. <\/span> [latex] 3f(1)-4g(-2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134086037\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134086039\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134086037-solution\">75<\/a><span class=\"os-divider\">. <\/span> [latex] f(\\frac{7}{3})-h(-2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165134373511\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<p id=\"fs-id1165135530586\">For the following exercises, graph [latex] y=x^2 [\/latex] on the given domain. Determine the corresponding range. Show each graph.<\/p>\r\n\r\n<div id=\"fs-id1165135530591\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135530593\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">76<\/span><span class=\"os-divider\">. <\/span> [latex] [-0.1, 0.1] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134087674\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134087676\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134087674-solution\">77<\/a><span class=\"os-divider\">. <\/span> [latex] [-10, 10] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135695182\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135695184\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">78<\/span><span class=\"os-divider\">. <\/span> [latex] [-100, 100] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135388446\">For the following exercises, graph [latex] y=x^3 [\/latex] on the given domain. Determine the corresponding range. Show each graph.<\/p>\r\n\r\n<div id=\"fs-id1165134060421\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134060424\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134060421-solution\">79<\/a><span class=\"os-divider\">. <\/span> [latex] [-0.1, 0.1] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133195224\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133195226\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">80<\/span><span class=\"os-divider\">. <\/span> [[latex] [-10, 10] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133402070\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133402072\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165133402070-solution\">81<\/a><span class=\"os-divider\">. <\/span> [latex] [-100, 100] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135634160\">For the following exercises, graph [latex] y=\\sqrt{x} [\/latex] on the given domain. Determine the corresponding range. Show each graph.<\/p>\r\n\r\n<div id=\"fs-id1165137844302\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137844304\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">82<\/span><span class=\"os-divider\">. <\/span> [latex] [0, 0.01] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137540730\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137540733\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137540730-solution\">83<\/a><span class=\"os-divider\">. <\/span> [latex] [0, 100] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137605840\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137605842\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">84<\/span><span class=\"os-divider\">. <\/span> [latex] [0, 10,000] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137539207\">For the following exercises, graph [latex] y=\\sqrt[3]{x} [\/latex] on the given domain. Determine the corresponding range. Show each graph.<\/p>\r\n\r\n<div id=\"fs-id1165134031227\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134031229\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134031227-solution\">85<\/a><span class=\"os-divider\">. <\/span> [latex] [-0.001, 0.001] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134087649\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134087651\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">86<\/span><span class=\"os-divider\">. <\/span> [latex] [-1000, 1000] [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135251229\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135251230\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135251229-solution\">87<\/a><span class=\"os-divider\">. <\/span> [latex] [-1,000,000, 1,000,000 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165135580349\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<div id=\"fs-id1165135580355\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135580357\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">88<\/span><span class=\"os-divider\">. <\/span>The amount of garbage, [latex] G [\/latex] produced by a city with population [latex] p [\/latex] is given by [latex] G=f(p). [\/latex] [latex] G [\/latex] is measured in tons per week, and [latex] p [\/latex] is measured in thousands of people.\r\n<div class=\"os-problem-container\">\r\n\r\n(a) The town of Windsor, Colorado, has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function [latex] f. [\/latex]\r\n\r\n(b) Explain the meaning of the statement [latex] f(5)=2. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137922382\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134269005\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137922382-solution\">89<\/a><span class=\"os-divider\">. <\/span>The number of cubic yards of dirt, [latex] D, [\/latex] needed to cover a garden with area [latex] a [\/latex] square feet is given by [latex] D=g(a). [\/latex]\r\n<div class=\"os-problem-container\">\r\n\r\n(a) A garden with area [latex] 5000\\hspace{0.5em}\\text{ft}^2 [\/latex] requires [latex] 50\\hspace{0.5em}\\text{yd}^3 [\/latex] of dirt. Express this information in terms of the function [latex] g. [\/latex]\r\n\r\n(b) Explain the meaning of the statement [latex] g(100)=1. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135553615\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135553617\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">90<\/span><span class=\"os-divider\">. <\/span>Let [latex] f(t) [\/latex] be the number of ducks in a lake [latex] t [\/latex] years after 1990. Explain the meaning of each statement:\r\n<div class=\"os-problem-container\">\r\n\r\n(a) [latex] f(5)=30 [\/latex]\r\n\r\n(b) [latex] f(10)=40 [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134272734\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134272736\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134272734-solution\">91<\/a><span class=\"os-divider\">. <\/span>Let [latex] h(t) [\/latex] be the height above ground, in feet, of a rocket [latex] t [\/latex] seconds after launching. Explain the meaning of each statement:\r\n<div class=\"os-problem-container\">\r\n\r\n(a) [latex] h(1)=200 [\/latex]\r\n\r\n(b) [latex] h(2)=350 [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135708043\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135708045\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">92<\/span><span class=\"os-divider\">. <\/span>Show that the function [latex] f(x)=3(x-5)^2+7 [\/latex] is <u>not<\/u>\u00a0one-to-one.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_55f2e8ec-a982-4586-9d48-a2f43d7b4107\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>In this section, you will:<\/p>\n<ul>\n<li>Determine whether a relation represents a function.<\/li>\n<li>Find the value of a function.<\/li>\n<li>Determine whether a function is one-to-one.<\/li>\n<li>Use the vertical line test to identify functions.<\/li>\n<li>Graph the functions listed in the library of functions.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137431376\">A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.<\/p>\n<section id=\"fs-id1165133394710\" data-depth=\"1\">\n<h2 data-type=\"title\">Determining Whether a Relation Represents a Function<\/h2>\n<p id=\"fs-id1165137781542\">A <span id=\"term-00005\" data-type=\"term\">relation<\/span> is a set of ordered pairs. The set of the first components of each <span id=\"term-00006\" class=\"no-emphasis\" data-type=\"term\">ordered pair<\/span> is called the <strong>domain <\/strong>and the set of the second components of each ordered pair is called the <strong>range<\/strong>. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.<\/p>\n<p style=\"text-align: center;\">[latex]\\{(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)\\}[\/latex]<\/p>\n<p id=\"fs-id1165133155834\">The domain is [latex]\\{1, 2, 3, 4, 5\\}.[\/latex] The range is [latex]\\{2, 4, 6, 8, 10\\}.[\/latex]<\/p>\n<p id=\"fs-id1165134234609\">Note that each value in the domain is also known as an <strong>input<\/strong> value, or <span id=\"term-00007\" data-type=\"term\">independent variable<\/span>, and is often labeled with the lowercase letter [latex]x.[\/latex] Each value in the range is also known as an <strong>output<\/strong> value, or <span id=\"term-00008\" data-type=\"term\">dependent variable<\/span>, and is often labeled lowercase letter [latex]y.[\/latex]<\/p>\n<p id=\"fs-id1165137748300\">A function [latex]f[\/latex] is a relation that assigns a single value in the range to each value in the domain<em data-effect=\"italics\">.<\/em> In other words, no <em data-effect=\"italics\">x<\/em>-values are repeated. For our example that relates the first five <span id=\"term-00009\" class=\"no-emphasis\" data-type=\"term\">natural numbers<\/span> to numbers double their values, this relation is a function because each element in the domain, [latex]\\{1, 2, 3, 4, 5\\},[\/latex] is paired with exactly one element in the range, [latex]\\{2, 4, 6, 8, 10\\}.[\/latex]<\/p>\n<p id=\"fs-id1165135421564\">Now let\u2019s consider the set of ordered pairs that relates the terms \u201ceven\u201d and \u201codd\u201d to the first five natural numbers. It would appear as<\/p>\n<p style=\"text-align: center;\">[latex]\\{(\\text{odd}, 1), (\\text{even}, 2), (\\text{odd}, 3), (\\text{even}, 4), (\\text{odd}, 5)\\}[\/latex]<\/p>\n<p id=\"fs-id1165135419796\">Notice that each element in the domain, [latex]\\{\\text{even}, \\text{odd}\\}[\/latex] is <em data-effect=\"italics\">not<\/em> paired with exactly one element in the range, [latex]\\{1, 2, 3, 4, 5\\}.[\/latex] For example, the term \u201codd\u201d corresponds to three values from the range, [latex]\\{1, 3, 5\\}[\/latex] and the term \u201ceven\u201d corresponds to two values from the range, [latex]\\{2, 4\\}[\/latex] This violates the definition of a function, so this relation is not a function.<\/p>\n<p id=\"fs-id1165135176295\">Figure 1 compares relations that are functions and not functions.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_449\" aria-describedby=\"caption-attachment-449\" style=\"width: 547px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-449\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-Fi.-1-300x90.jpg\" alt=\"\" width=\"547\" height=\"164\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-Fi.-1-300x90.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-Fi.-1-768x231.jpg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-Fi.-1-65x20.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-Fi.-1-225x68.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-Fi.-1-350x105.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-Fi.-1.jpg 943w\" sizes=\"auto, (max-width: 547px) 100vw, 547px\" \/><figcaption id=\"caption-attachment-449\" class=\"wp-caption-text\">Three relations that demonstrate what constitute a function.<br \/>Figure 1 (a) This relationship is a function because each input is associated with a single output. Note that input q and r both give output n. (b) This relationship is also a function. In this case, each input is associated with a single output. (c) This relationship is not a function because input q is associated with two different outputs.<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A <strong>function<\/strong> is a relation in which each possible input value leads to exactly one output value. We say &#8220;the output is a function of the input.&#8221;<\/p>\n<p>The <strong>input<\/strong> values make up the\u00a0<strong>domain<\/strong>, and the <strong>output<\/strong> values make up the\u00a0<strong>range<\/strong>.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given a relationship between two quantities, determine whether the relationship is a function.<\/p>\n<ol>\n<li>Identify the input values.<\/li>\n<li>Identify the output values.<\/li>\n<li>If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example I: Determining in Menu Price Lists are Functions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A donut shop in Park Meadows is a local favorite. Their menu, shown below, consists of items and their prices.<\/p>\n<p>(a) Is price a function of the item?<\/p>\n<p>(b) Is the item a function of the price?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-450 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1-300x144.jpg\" alt=\"\" width=\"479\" height=\"230\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1-300x144.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1-65x31.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1-225x108.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1-350x168.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1.jpg 584w\" sizes=\"auto, (max-width: 479px) 100vw, 479px\" \/><\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a) Let&#8217;s begin by considering the input as the items on the menu. The output values are then the prices.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1149 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1-2-300x144.jpg\" alt=\"\" width=\"519\" height=\"249\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1-2-300x144.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1-2-65x31.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1-2-225x108.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1-2-350x168.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1-2.jpg 584w\" sizes=\"auto, (max-width: 519px) 100vw, 519px\" \/><\/p>\n<p>Each item on the menu has only one price, so the price is a function of the item.<\/p>\n<p>(b) Two items on the menu have the same price. If we consider the prices to be the input values and the items to be the output, then the same input value could have more than one output associated with it. See the image below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-452 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1.1-300x144.jpg\" alt=\"\" width=\"523\" height=\"251\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1.1-300x144.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1.1-65x31.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1.1-225x108.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1.1-350x168.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-1.1.jpg 584w\" sizes=\"auto, (max-width: 523px) 100vw, 523px\" \/><\/p>\n<p>Therefore, the item is not a function of the price.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2: Determining if Class Grade Rules are Functions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>In a College Algebra class, the overall percent grade corresponds to a grade point average. Is grade point average a function of the percent grade? Is the percent grade a function of the grade point average? Table 1 shows a possible rule for assigning grade points.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 1<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 11.1111%;\"><strong>Percent grade<\/strong><\/td>\n<td style=\"width: 11.1111%;\">0-56<\/td>\n<td style=\"width: 11.1111%;\">57-61<\/td>\n<td style=\"width: 11.1111%;\">62-66<\/td>\n<td style=\"width: 11.1111%;\">67-71<\/td>\n<td style=\"width: 11.1111%;\">72-77<\/td>\n<td style=\"width: 11.1111%;\">78-86<\/td>\n<td style=\"width: 11.1111%;\">87-91<\/td>\n<td style=\"width: 11.1111%;\">92-100<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 11.1111%;\"><strong>Grade point average<\/strong><\/td>\n<td style=\"width: 11.1111%;\">0.0<\/td>\n<td style=\"width: 11.1111%;\">1.0<\/td>\n<td style=\"width: 11.1111%;\">1.5<\/td>\n<td style=\"width: 11.1111%;\">2.0<\/td>\n<td style=\"width: 11.1111%;\">2.5<\/td>\n<td style=\"width: 11.1111%;\">3.0<\/td>\n<td style=\"width: 11.1111%;\">3.5<\/td>\n<td style=\"width: 11.1111%;\">4.0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>For any percent grade earned, there is an associated grade point average, so the grade point average is a function of the percent grade. In other words, if we input the percent grade, the output is a specific grade point average.<\/p>\n<p>In the grading system given, there is a range of percent grades that correspond to the same grade point average. For example, students who receive a grade point average of 3.0 could have a variety of percent grades ranging from 78 all the way to 86. Thus, percent grade is not a function of grade point average.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Table 2 lists five students in a mathematics contest at CCA in order of rank.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Tab;e 2<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><strong>Student<\/strong><\/td>\n<td style=\"width: 50%;\"><strong>Rank<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Daniela<\/td>\n<td style=\"width: 50%;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Amir<\/td>\n<td style=\"width: 50%;\">2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">John<\/td>\n<td style=\"width: 50%;\">3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Mary<\/td>\n<td style=\"width: 50%;\">4<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Diego<\/td>\n<td style=\"width: 50%;\">5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>(a) Is the rank a function of the student name?<\/p>\n<p>(b) Is the student name a function of the rank?<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165134474160\" data-depth=\"2\">\n<h3 data-type=\"title\">Using Function Notation<\/h3>\n<p id=\"fs-id1165133359348\">Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard <span id=\"term-00015\" class=\"no-emphasis\" data-type=\"term\">function notation<\/span> is one representation that facilitates working with functions.<\/p>\n<p id=\"fs-id1165137453971\">To represent \u201cheight is a function of age,\u201d we start by identifying the descriptive variables [latex]h[\/latex] for height and [latex]a[\/latex] for age. The letters [latex]f, g[\/latex] and [latex]h[\/latex] are often used to represent functions just as we use [latex]x, y[\/latex] and [latex]z[\/latex] to represent numbers and [latex]A, B[\/latex] and [latex]C[\/latex] to represent sets.<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{lllll}h \\; \\text{is} \\; f \\; \\text{of} \\; a &&&&\\text{We name the function } f; \\; \\text{height is a function of age.} \\\\h = f(a) &&&&\\text{We use parentheses to indicate the function input.} \\\\f(a) &&&&\\text{We name the function } f; \\; \\text{the expression is read as ``}f \\; \\text{of } a\\text{.\"} \\\\\\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165137766965\">Remember, we can use any letter to name the function; the notation [latex]h(a)[\/latex] shows us that [latex]h[\/latex] depends on [latex]a.[\/latex] The value [latex]a[\/latex] must be put into the function [latex]h[\/latex] to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.<\/p>\n<p id=\"fs-id1165135436660\">We can also give an algebraic expression as the input to a function. For example [latex]f(a+b)[\/latex] means \u201cfirst add <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em>, and the result is the input for the function <em data-effect=\"italics\">f<\/em>.\u201d The operations must be performed in this order to obtain the correct result.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Function Notation<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The notation\u00a0[latex]y=f(x)[\/latex] defines a function named\u00a0[latex]f.[\/latex] This is read as &#8220;[latex]y[\/latex] is a function of [latex]x.[\/latex]&#8221; The letter\u00a0[latex]x[\/latex] represents the input value, or independent variable. The letter\u00a0[latex]y[\/latex] or\u00a0[latex]f(x)[\/latex] represents the output value, or dependent variable.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Using Function Notation for Days in a Month<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use function notation to represent a function whose input is the name of a month and output is the number of days in that month. Assume that the domain does not include leap years.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The number of days in a month is a function of the name of the month, so if we name the function\u00a0[latex]f,[\/latex] we write\u00a0[latex]\\text{days}=f(\\text{month})[\/latex] or [latex]d=f(m)[\/latex] The name of the month is the input to a &#8220;rule&#8221; that associates a specific number (the output) with each input.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_455\" aria-describedby=\"caption-attachment-455\" style=\"width: 354px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-455\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-3.jpg\" alt=\"\" width=\"354\" height=\"205\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-3.jpg 245w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-3-65x38.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-3-225x130.jpg 225w\" sizes=\"auto, (max-width: 354px) 100vw, 354px\" \/><figcaption id=\"caption-attachment-455\" class=\"wp-caption-text\">Figure 2<\/figcaption><\/figure>\n<p>For example,\u00a0[latex]f(\\text{March})=31[\/latex] because March has 31 days. The notation\u00a0[latex]d=f(m)[\/latex] reminds us that the number of days, [latex]d[\/latex](the output), is dependent on the name of the month, [latex]m[\/latex](the input).<\/p>\n<h3>Analysis<\/h3>\n<p>Note that the inputs of a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4: Interpreting Function Notation<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A function\u00a0[latex]N=f(y)[\/latex] gives the numbers of police officers, [latex]N[\/latex] in Aurora, Colorado, in year [latex]y.[\/latex] What does\u00a0[latex]f(2005)=300[\/latex] represent?<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>When we read [latex]f(2005)=300,[\/latex], we see that the input year is 2005. The value for the output, the number of police officers\u00a0[latex](N),[\/latex] is 300. Remember,\u00a0[latex]N=f(y).[\/latex] The statement\u00a0[latex]f(2005)=300[\/latex] tells us that in the year 2005 there were 300 police officers in Aurora.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Use function notation to express the weight of a pig in pounds as a function of its age in days [latex]d.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Q: Instead of notation such as\u00a0[latex]y=f(x),[\/latex] could we use the same symbol for the output as for the function, such as\u00a0[latex]y=y(x),[\/latex] meaning &#8220;[latex]y[\/latex] is a function of [latex]x[\/latex]?&#8221;<\/p>\n<p><em>A: Yes, this is often done, especially in applied subjects that use higher math, such as physics and engineering. However, in exploring math itself we like to maintain a distinction between a function such as\u00a0[latex]f[\/latex] which is a rule or procedure, and the output\u00a0[latex]y[\/latex] we get by applying\u00a0[latex]f[\/latex] to a particular input\u00a0[latex]x.[\/latex] This is why we usually use notation such as\u00a0[latex]y=f(x), P=W(d),[\/latex] and so on.<\/em><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137804204\" data-depth=\"2\">\n<h3 data-type=\"title\">Representing Functions Using Tables<\/h3>\n<p id=\"fs-id1165137648317\">A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values.\u00a0In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship.<\/p>\n<p id=\"fs-id1165137761188\">Table 3 lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function\u00a0[latex]f[\/latex] where [latex]D=f(m)[\/latex] identifies months by an integer rather than by name.<\/p>\n<div id=\"Table_01_01_03\" class=\"os-table\">\n<table class=\"grid\" data-id=\"Table_01_01_03\">\n<caption>Table 3<\/caption>\n<tbody>\n<tr>\n<td data-align=\"center\"><strong>Month number, [latex]m[\/latex] <\/strong>(input)<\/td>\n<td data-align=\"center\">1<\/td>\n<td data-align=\"center\">2<\/td>\n<td data-align=\"center\">3<\/td>\n<td data-align=\"center\">4<\/td>\n<td data-align=\"center\">5<\/td>\n<td data-align=\"center\">6<\/td>\n<td data-align=\"center\">7<\/td>\n<td data-align=\"center\">8<\/td>\n<td data-align=\"center\">9<\/td>\n<td data-align=\"center\">10<\/td>\n<td data-align=\"center\">11<\/td>\n<td data-align=\"center\">12<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\"><strong>Days in month, [latex]D[\/latex] <\/strong>(output)<\/td>\n<td data-align=\"center\">31<\/td>\n<td data-align=\"center\">28<\/td>\n<td data-align=\"center\">31<\/td>\n<td data-align=\"center\">30<\/td>\n<td data-align=\"center\">31<\/td>\n<td data-align=\"center\">30<\/td>\n<td data-align=\"center\">31<\/td>\n<td data-align=\"center\">31<\/td>\n<td data-align=\"center\">30<\/td>\n<td data-align=\"center\">31<\/td>\n<td data-align=\"center\">30<\/td>\n<td data-align=\"center\">31<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\">\n<p>Table 4 defines a function [latex]Q=g(n).[\/latex] Remember, this notation tells us that [latex]g[\/latex] is the name of the function that takes the input [latex]n[\/latex] and gives the output [latex]Q.[\/latex]<\/p>\n<div id=\"Table_01_01_04\" class=\"os-table\">\n<table class=\"grid\" data-id=\"Table_01_01_04\">\n<caption>Table 4<\/caption>\n<colgroup>\n<col \/>\n<col data-width=\"25\" \/>\n<col data-width=\"25\" \/>\n<col data-width=\"25\" \/>\n<col data-width=\"25\" \/>\n<col data-width=\"25\" \/><\/colgroup>\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]n[\/latex]<\/td>\n<td data-align=\"center\">1<\/td>\n<td data-align=\"center\">2<\/td>\n<td data-align=\"center\">3<\/td>\n<td data-align=\"center\">4<\/td>\n<td data-align=\"center\">5<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]Q[\/latex]<\/td>\n<td data-align=\"center\">8<\/td>\n<td data-align=\"center\">6<\/td>\n<td data-align=\"center\">7<\/td>\n<td data-align=\"center\">6<\/td>\n<td data-align=\"center\">8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\">Table 5 displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.<\/div>\n<\/div>\n<div id=\"Table_01_01_05\" class=\"os-table\">\n<table class=\"grid\" data-id=\"Table_01_01_05\">\n<caption>Table 5<\/caption>\n<colgroup>\n<col \/>\n<col data-width=\"30\" \/>\n<col data-width=\"30\" \/>\n<col data-width=\"30\" \/>\n<col data-width=\"30\" \/>\n<col data-width=\"30\" \/>\n<col data-width=\"30\" \/>\n<col data-width=\"30\" \/><\/colgroup>\n<tbody>\n<tr>\n<td data-align=\"center\"><strong>Age in years, <\/strong> [latex]a[\/latex] (input)<\/td>\n<td data-align=\"center\">5<\/td>\n<td data-align=\"center\">5<\/td>\n<td data-align=\"center\">6<\/td>\n<td data-align=\"center\">7<\/td>\n<td data-align=\"center\">8<\/td>\n<td data-align=\"center\">9<\/td>\n<td data-align=\"center\">10<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\"><strong>Height in inches, <\/strong> [latex]h[\/latex] (output)<\/td>\n<td data-align=\"center\">40<\/td>\n<td data-align=\"center\">42<\/td>\n<td data-align=\"center\">44<\/td>\n<td data-align=\"center\">47<\/td>\n<td data-align=\"center\">50<\/td>\n<td data-align=\"center\">52<\/td>\n<td data-align=\"center\">54<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><\/div>\n<div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a table of input and output values, determine whether the table represents a function.<\/strong><\/p>\n<ol>\n<li>Identify the input and output values.<\/li>\n<li>Check to see if each input value is paired with only one output value. If so, the table represents a function.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 5: Identifying Tables that Represent Functions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Which table (if any), Table 6, Table 7, or Table 8, represents a function?<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 6<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 50%; text-align: center;\"><strong>Input<\/strong><\/td>\n<td style=\"width: 50%; text-align: center;\"><strong>Output<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">2<\/td>\n<td style=\"width: 50%; text-align: center;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">5<\/td>\n<td style=\"width: 50%; text-align: center;\">3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">8<\/td>\n<td style=\"width: 50%; text-align: center;\">6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 7<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 50%; text-align: center;\"><strong>Input<\/strong><\/td>\n<td style=\"width: 50%; text-align: center;\"><strong>Output<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">-3<\/td>\n<td style=\"width: 50%; text-align: center;\">5<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">0<\/td>\n<td style=\"width: 50%; text-align: center;\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">4<\/td>\n<td style=\"width: 50%; text-align: center;\">5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 8<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 50%; text-align: center;\"><strong>Input<\/strong><\/td>\n<td style=\"width: 50%; text-align: center;\"><strong>Output<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">1<\/td>\n<td style=\"width: 50%; text-align: center;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">5<\/td>\n<td style=\"width: 50%; text-align: center;\">2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">5<\/td>\n<td style=\"width: 50%; text-align: center;\">4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Table 6 and Table 7 define functions. In both, each input value corresponds to exactly one output value. Table 8 does not define a function because the input value of 5 corresponds to two different output values.<\/p>\n<p>When a table represents a function, corresponding input and output values can also be specified using function notation.<\/p>\n<p>The function represented by Table 6 can be represented by writing<\/p>\n<p style=\"text-align: center;\">[latex]f(2) = 1, f(5) = 3, \\text{and } f(8) = 6[\/latex]<\/p>\n<p>Similarly, the statements<\/p>\n<p style=\"text-align: center;\">[latex]g(-3) = 5, \\; g(0) = 1, \\text{and } g(4) = 5[\/latex]<\/p>\n<p>represent the function in Table 7.<\/p>\n<p>Table 8 cannot be expressed in a similar way because it does not represent a function.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Does Table 9 represent a function? Why or why not?<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 9<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 50%; text-align: center;\"><strong>Input<\/strong><\/td>\n<td style=\"width: 50%; text-align: center;\"><strong>Output<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">1<\/td>\n<td style=\"width: 50%; text-align: center;\">10<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">2<\/td>\n<td style=\"width: 50%; text-align: center;\">100<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%; text-align: center;\">3<\/td>\n<td style=\"width: 50%; text-align: center;\">1000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<section id=\"fs-id1165137503241\" data-depth=\"1\">\n<h2 data-type=\"title\">Finding Input and Output Values of a Function<\/h2>\n<p id=\"fs-id1165137470651\">When we know an input value and want to determine the corresponding output value for a function, we <em data-effect=\"italics\">evaluate<\/em> the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.<\/p>\n<p id=\"fs-id1165137735634\">When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function\u2019s formula and <em data-effect=\"italics\">solve<\/em> for the input. Solving can produce more than one solution because different input values can produce the same output value.<\/p>\n<section id=\"fs-id1165137425943\" data-depth=\"2\">\n<h3 data-type=\"title\">Evaluation of Functions in Algebraic Forms<\/h3>\n<p id=\"fs-id1165137655584\">When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function [latex]f(x)=5-3x^2[\/latex] can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given the formula for a function, evaluate.<\/strong><\/p>\n<ol>\n<li>Substitute the input variable in the formula for the value provided.<\/li>\n<li>Calculate the result.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 6: Evaluating Functions at Specific Values<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Evaluate\u00a0[latex]f(x)=x^2+3x-4[\/latex] at:<\/p>\n<p>(a) [latex]2[\/latex]<\/p>\n<p>(b) [latex]a[\/latex]<\/p>\n<p>(c) [latex]a+h[\/latex]<\/p>\n<p>(d) Now evaluate [latex]\\frac{f(a+h)-f(a)}{h}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Replace the\u00a0[latex]x[\/latex] in the function with each specified value.<\/p>\n<p>(a) Because the input value is a number 2, we can use simple algebra to simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{ll} f(2) & =2^2+3(2)-4 \\\\ & = 4+6-4 \\\\ & = 6 \\end{array}[\/latex]<\/p>\n<p>(b) In this case, the input value is a letter so we cannot simplify the answer any further.<\/p>\n<p style=\"text-align: center;\">[latex]f(a)=a^2+3a-4[\/latex]<\/p>\n<p>(c) With an input value of [latex]a+h[\/latex], we must use the distributive property.<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{ll} f(a+h) & =(a+h)^2+3(a+h)-4 \\\\ & = a^2+2ah+h^2+3a+3h-4 \\end{array}[\/latex]<\/p>\n<p>(d) In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that<\/p>\n<p style=\"text-align: center;\">[latex]f(a+h)=a^2+2ah+h^2+3a+3h-4[\/latex]<\/p>\n<p id=\"fs-id1165135632109\">and we know that<\/p>\n<p style=\"text-align: center;\">[latex]f(a)=a^2+3a-4[\/latex]<\/p>\n<p id=\"fs-id1165137767461\">Now we combine the results and simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{lll} \\frac{f(a+h)-f(a)}{h} & =\\frac{(a^2+2ah+h^2+3a+3h-4)+(a^2+3a-4)}{h} \\\\ & = \\frac{2ah+h^2+3h}{h} \\\\ & = \\frac{h(2a+h+3)}{h} & \\text{Factor out} \\ h. \\\\ & = 2a+h+3 & \\text{Simplify.} \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 7: Evaluating Functions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given the function\u00a0[latex]h(p)=p^2+2p,[\/latex] evaluate [latex]h(4).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>To evaluate [latex]h(4),[\/latex] we substitute the value 4 for the input variable\u00a0[latex]p[\/latex] in the given function.<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{ll} h(p) & =p^2+2p \\\\ h(4) & =(4^2)+2(4) \\\\ & =16+8 \\\\ & =24 \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165137785006\">Therefore, for an input of 4, we have an output of 24.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given the function [latex]g(m)=\\sqrt{m-4},[\/latex] evaluate [latex]g(5).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 8: Solving Functions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given the function\u00a0[latex]h(p)=p^2+2p,[\/latex] solve for [latex]h(p)=3.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{lll} h(p) & =3 \\\\ p^2+2p & =3 & \\text{Substitute the original function} \\ h(p)=p^2+2p. \\\\ p^2+2p-3 & =0 & \\text{Subtract 3 from each side.} \\\\ (p+3)(p-1) & =0 & \\text{Factor.} \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165137770370\">If [latex](p+3)(p-1)=0,[\/latex] either [latex](p+3)=0[\/latex] or [latex](p-1)=0[\/latex] (or both of them equal 0). We will set each factor equal to 0 and solve for in each case.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll} (p+3) & =0, & p=-3 \\\\ (p-1) & =0, & p=1 \\end{array}[\/latex]<\/p>\n<p>This gives us two solutions. The output\u00a0[latex]h(p)=3[\/latex] when the input is either\u00a0[latex]p=1[\/latex] or [latex]p=-3[\/latex] We can also verify by graphing as in Figure 3. The graph verifies that [latex]h(1)=h(-3)=3[\/latex]\u00a0and [latex]h(4)=24.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_457\" aria-describedby=\"caption-attachment-457\" style=\"width: 465px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-457\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-8-294x300.jpg\" alt=\"\" width=\"465\" height=\"474\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-8-294x300.jpg 294w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-8-65x66.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-8-225x229.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-8-350x357.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-ex-8.jpg 359w\" sizes=\"auto, (max-width: 465px) 100vw, 465px\" \/><figcaption id=\"caption-attachment-457\" class=\"wp-caption-text\">Figure 3<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given the function [latex]g(m)=\\sqrt{m-4},[\/latex] solve [latex]g(m)=2.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137591827\" data-depth=\"2\">\n<h3 data-type=\"title\">Evaluating Functions Expressed in Formulas<\/h3>\n<p id=\"fs-id1165137598337\">Some functions are defined by mathematical rules or procedures expressed in <span id=\"term-00016\" class=\"no-emphasis\" data-type=\"term\">equation<\/span> form. If it is possible to express the function output with a <span id=\"term-00017\" class=\"no-emphasis\" data-type=\"term\">formula<\/span> involving the input quantity, then we can define a function in algebraic form. For example, the equation\u00a0[latex]2n+6p=12[\/latex] expresses a functional relationship between [latex]n[\/latex] and [latex]p.[\/latex] We can rewrite it to decide if [latex]p[\/latex] is a function of [latex]n.[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a function in equation form, write is algebraic formula.<\/strong><\/p>\n<ol>\n<li>Solve the equation to isolate the output variable on one side of the equal sign, with the other side as an expression that involves <em>only<\/em> the input variable.<\/li>\n<li>Use all the usual algebraic methods for solving equations, such as adding or subtracting the same quantity to or from both sides, or multiplying or dividing both sides of the equation by the same quantity.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 9: Finding the Algebraic Form of a Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Express the relationship\u00a0[latex]2n+6p=12[\/latex] as a function\u00a0[latex]p=f(n),[\/latex] if possible.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>To express the relationship in this form, we need to be able to write the relationship where\u00a0[latex]p[\/latex] is a function of\u00a0[latex]n,[\/latex] which means writing it as [latex]p=[\\text{expression involving}\\hspace{0.25em}n].[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{lll} 2n+6p & =12 \\\\ 6p & =12-2n & \\text{Subtract} \\ 2n \\ \\text{from both sides.} \\\\ p & =\\frac{12-2n}{6} & \\text{Divide both sides by 6 and simplify.} \\\\ p &=\\frac{12}{6}-\\frac{2n}{6} \\\\ p & = 2-\\frac{1}{3}n \\end{array}[\/latex]<\/p>\n<p>Therefore, [latex]p[\/latex] as a function of [latex]n[\/latex] is written as<\/p>\n<p>[latex]\\hspace{3em}p=f(n)=2-\\frac{1}{3}n[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 10: Expressing the Equation of a Circle as a Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Does the equation\u00a0[latex]x^2+y^2=1[\/latex] represent a function with\u00a0[latex]x[\/latex] as input and\u00a0[latex]y[\/latex] as output? If so, express the relationship as a function [latex]y=f(x).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>First we subtract\u00a0[latex]x^2[\/latex] from both sides.<\/p>\n<p style=\"text-align: center;\">[latex]y^2=1-x^2[\/latex]<\/p>\n<p>We now try to solve [latex]y[\/latex] for this equation.<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{ll} y & =\\pm\\sqrt{1-x^2} \\\\ & =+\\sqrt{1-x^2} \\ \\text{and} \\ -\\sqrt{1-x^2} \\end{array}[\/latex]<\/p>\n<p>We get two outputs corresponding to the same input, so this relationship cannot be represented as a single function [latex]y=f(x).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"body\">\n<div id=\"fs-id1165135378843\" class=\"unnumbered\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165135378845\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>If\u00a0[latex]x=8y^3=0,[\/latex] express\u00a0[latex]y[\/latex] as a function of [latex]x.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Q: Are there relationships expressed by an equation that do represent a function but which still cannot be represented by an algebraic formula?<\/strong><\/p>\n<p><em>A: Yes, this can happen. For example, given the equation\u00a0[latex]x=y+2^y,[\/latex] if we want to express [latex]y[\/latex] as a function of [latex]x,[\/latex] there is no simple algebraic formula involving only\u00a0[latex]x[\/latex] that equals\u00a0[latex]y.[\/latex] However, each\u00a0[latex]x[\/latex] does determine a unique value for\u00a0[latex]y,[\/latex] and there are mathematical procedures by which [latex]y[\/latex] can be found to any desired accuracy. In this case, we say that the equation gives an implicit (implied) rule for\u00a0[latex]y[\/latex] as a function of\u00a0[latex]x,[\/latex] even though the formula cannot be written explicitly.<\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<section id=\"fs-id1165137648450\" data-depth=\"2\">\n<h3 data-type=\"title\">Evaluating a Function Given in Tabular Form<\/h3>\n<p id=\"fs-id1165135186424\">As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy\u2019s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.<\/p>\n<p id=\"fs-id1165135186427\" class=\"has-noteref\">The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See Table 10.<a class=\"footnote\" title=\"http:\/\/www.kgbanswers.com\/how-long-is-a-dogs-memory-span\/4221590. Accessed 3\/24\/2014.\" id=\"return-footnote-132-1\" href=\"#footnote-132-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><sup id=\"footnote-ref2\" data-type=\"footnote-number\"><\/sup><\/p>\n<div id=\"Table_01_01_10\" class=\"os-table\">\n<table class=\"grid\" data-id=\"Table_01_01_10\">\n<caption>Table 10<\/caption>\n<colgroup>\n<col data-width=\"85\" data-align=\"center\" \/>\n<col data-align=\"center\" \/><\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: center;\" scope=\"col\" data-align=\"center\">Pet<\/th>\n<th style=\"text-align: center;\" scope=\"col\" data-align=\"center\">Memory span in hours<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center;\" data-align=\"center\">Puppy<\/td>\n<td style=\"text-align: center;\" data-align=\"center\">0.008<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" data-align=\"center\">Adult dog<\/td>\n<td style=\"text-align: center;\" data-align=\"center\">0.083<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" data-align=\"center\">Cat<\/td>\n<td style=\"text-align: center;\" data-align=\"center\">16<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" data-align=\"center\">Goldfish<\/td>\n<td style=\"text-align: center;\" data-align=\"center\">2160<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" data-align=\"center\">Beta fish<\/td>\n<td style=\"text-align: center;\" data-align=\"center\">3600<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\">At times, evaluating a function in table form may be more useful than using equations. Here let us call the function\u00a0[latex]P.[\/latex] The domain of the function is the type of pet and the range is a real number representing the number of hours the pet\u2019s memory span lasts. We can evaluate the function [latex]P[\/latex] at the input value of \u201cgoldfish.\u201d We would write [latex]P(\\text{goldfish})=2160.[\/latex] Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function [latex]P[\/latex] seems ideally suited to this function, more so than writing it in paragraph or function form.<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a function represented by a table, identify specific output and input values.<\/strong><\/p>\n<ol>\n<li>Find the given input in the row (or column) of input values.<\/li>\n<li>Identify the corresponding output value paired with that input value.<\/li>\n<li>Find the given output values in the row (or column) of output values, noting every time that output value appears.<\/li>\n<li>Identify the input value(s) corresponding to the given output value.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 11: Evaluating and Solving a Tabular Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Using Table 11,<\/p>\n<p>(a) Evaluate [latex]g(3).[\/latex]<\/p>\n<p>(b) Solve [latex]g(n)-6.[\/latex]<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 11<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 16.6667%;\">[latex]n[\/latex]<\/td>\n<td style=\"width: 16.6667%;\">1<\/td>\n<td style=\"width: 16.6667%;\">2<\/td>\n<td style=\"width: 16.6667%;\">3<\/td>\n<td style=\"width: 16.6667%;\">4<\/td>\n<td style=\"width: 16.6667%;\">5<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;\">[latex]g(n)[\/latex]<\/td>\n<td style=\"width: 16.6667%;\">8<\/td>\n<td style=\"width: 16.6667%;\">6<\/td>\n<td style=\"width: 16.6667%;\">7<\/td>\n<td style=\"width: 16.6667%;\">6<\/td>\n<td style=\"width: 16.6667%;\">8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a) Evaluating\u00a0[latex]g(3)[\/latex] means determining the output value of the function\u00a0[latex]g[\/latex] for the input value of\u00a0[latex]n=3.[\/latex] The table output value corresponding to\u00a0[latex]n=3[\/latex] is 7, so [latex]g(3)=7.[\/latex]<\/p>\n<p>(b) Solving\u00a0[latex]g(n)=6[\/latex] means identifying the input values,\u00a0[latex]n,[\/latex] that produce an output of 6. The table shows two solutions:\u00a0[latex]2[\/latex] and [latex]4.[\/latex]<\/p>\n<table class=\"grid\">\n<tbody>\n<tr>\n<td style=\"width: 171px;\">[latex]n[\/latex]<\/td>\n<td style=\"width: 97px;\">1<\/td>\n<td style=\"width: 98px;\">2<\/td>\n<td style=\"width: 97px;\">3<\/td>\n<td style=\"width: 98px;\">4<\/td>\n<td style=\"width: 97px;\">5<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 171px;\">[latex]g(n)[\/latex]<\/td>\n<td style=\"width: 97px;\">8<\/td>\n<td style=\"width: 98px;\">6<\/td>\n<td style=\"width: 97px;\">7<\/td>\n<td style=\"width: 98px;\">6<\/td>\n<td style=\"width: 97px;\">8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When we input 2 into the function\u00a0[latex]g,[\/latex] our output is 6. When we input 4 into the function\u00a0[latex]g,[\/latex] our output is also 6.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Using the table from <strong>Example 11: Evaluating and Solving a Tabular Function<\/strong> above, evaluate [latex]g(1).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165135696152\" data-depth=\"2\">\n<h3 data-type=\"title\">Finding Function Values from a Graph<\/h3>\n<p id=\"fs-id1165137779152\">Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 12: Reading Function Values from a Graph<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given the graph in Figure 4,<\/p>\n<p>(a) Evaluate [latex]f(2).[\/latex]<\/p>\n<p>(b) Solve [latex]f(x)=4.[\/latex]<\/p>\n<figure id=\"attachment_459\" aria-describedby=\"caption-attachment-459\" style=\"width: 505px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-459\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-4-300x283.jpg\" alt=\"\" width=\"505\" height=\"476\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-4-300x283.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-4-65x61.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-4-225x212.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-4-350x330.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-4.jpg 361w\" sizes=\"auto, (max-width: 505px) 100vw, 505px\" \/><figcaption id=\"caption-attachment-459\" class=\"wp-caption-text\">Figure 4<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a) To evaluate\u00a0[latex]f(2),[\/latex] locate the point on the curve where\u00a0[latex]x=2,[\/latex] then read the y-coordinate of that point. The point has coordinates\u00a0[latex](2, 1),[\/latex] so\u00a0[latex]f(2)=1.[\/latex] See Figure 5.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_460\" aria-describedby=\"caption-attachment-460\" style=\"width: 505px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-460\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-5-300x282.jpg\" alt=\"\" width=\"505\" height=\"475\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-5-300x282.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-5-65x61.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-5-225x211.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-5-350x329.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-5.jpg 361w\" sizes=\"auto, (max-width: 505px) 100vw, 505px\" \/><figcaption id=\"caption-attachment-460\" class=\"wp-caption-text\">Figure 5<\/figcaption><\/figure>\n<p>(b) To solve\u00a0[latex]f(x)=4,[\/latex] we find the output value [latex]4[\/latex] on the vertical axis. Moving horizontally along the line\u00a0[latex]y=4,[\/latex] we locate two points of the curve with output value [latex]4: (-1, 4)[\/latex] and\u00a0[latex](3, 4).[\/latex] These points represent the two solutions to\u00a0[latex]f(x)=4: -1[\/latex] or\u00a0[latex]3.[\/latex] This means\u00a0[latex]f(-1)=4[\/latex] and [latex]- f(3)=4[\/latex] or when the input is\u00a0[latex]-1[\/latex] or\u00a0[latex]3[\/latex] the output is\u00a0[latex]4.[\/latex] See Figure 6.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_461\" aria-describedby=\"caption-attachment-461\" style=\"width: 491px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-461\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-6-296x300.jpg\" alt=\"\" width=\"491\" height=\"498\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-6-296x300.jpg 296w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-6-65x66.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-6-225x228.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-6-350x355.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-6.jpg 361w\" sizes=\"auto, (max-width: 491px) 100vw, 491px\" \/><figcaption id=\"caption-attachment-461\" class=\"wp-caption-text\">Figure 6<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Using Figure 4 above, solve [latex]f(x)=1.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135422920\" data-depth=\"1\">\n<h2 data-type=\"title\">Determining Whether a Function is One-to-One<\/h2>\n<div>\n<div>\n<p>Some functions have a given output value that corresponds to two or more input values. For example, in the case chart shown in the figure at the beginning of this chapter, the world-wide cases were 7,000,000 for three different weeks, meaning that there were three different input values that all resulted in the same output value of 7,000,000.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135245630\">However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in <a class=\"autogenerated-content\" href=\"3-1-functions-and-function-notation#Table_01_01_13\">Table 12<\/a>.<\/p>\n<div id=\"Table_01_01_13\" class=\"os-table\">\n<table class=\"grid\" style=\"height: 70px;\" data-id=\"Table_01_01_13\">\n<caption>Table 12<\/caption>\n<colgroup>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/><\/colgroup>\n<thead>\n<tr style=\"height: 15px;\">\n<th style=\"height: 15px; width: 258.083px; text-align: center;\" scope=\"col\" data-align=\"center\">Letter grade<\/th>\n<th style=\"height: 15px; width: 402.483px; text-align: center;\" scope=\"col\" data-align=\"center\">Grade point average<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 258.083px; text-align: center;\" data-align=\"center\">A<\/td>\n<td style=\"height: 15px; width: 402.483px; text-align: center;\" data-align=\"center\">4.0<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 258.083px; text-align: center;\" data-align=\"center\">B<\/td>\n<td style=\"height: 15px; width: 402.483px; text-align: center;\" data-align=\"center\">3.0<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; width: 258.083px; text-align: center;\" data-align=\"center\">C<\/td>\n<td style=\"height: 15px; width: 402.483px; text-align: center;\" data-align=\"center\">2.0<\/td>\n<\/tr>\n<tr style=\"height: 10px;\">\n<td style=\"height: 10px; width: 258.083px; text-align: center;\" data-align=\"center\">D<\/td>\n<td style=\"height: 10px; width: 402.483px; text-align: center;\" data-align=\"center\">1.0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\">This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.<\/div>\n<\/div>\n<p id=\"fs-id1165137628999\">To visualize this concept, let\u2019s look again at the two simple functions sketched in Figure 1<strong>(a) <\/strong>and Figure 1<strong>(b)<\/strong>. The function in part (a) shows a relationship that is not a one-to-one function because inputs\u00a0[latex]q[\/latex] and [latex]r[\/latex] both give output [latex]n.[\/latex] The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">One-to-One Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A <strong>one-to-one function<\/strong> is a function in which each output value corresponds to exactly one input value.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 13: Determining Whether a Relationship is a One-to-One Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Is the area of a circle a function of its radius? If yes, is the function one-to-one?<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>A circle of radius\u00a0[latex]r[\/latex] has a unique area measure given by\u00a0[latex]A=\\pi r^2[\/latex] so for any input [latex]r,[\/latex] there is only one output,\u00a0[latex]A.[\/latex] The area is a function of radius [latex]r.[\/latex]<\/p>\n<p>If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure\u00a0[latex]A[\/latex] is given by the formula [latex]A=\\pi r^2.[\/latex]\u00a0Because areas and radii are positive numbers, there is exactly one solution:\u00a0[latex]\\sqrt{\\frac{A}{\\pi}}.[\/latex] So the area of a circle is a one-to-one function of the circle&#8217;s radius.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>(a) Is a balance a function of a bank account number?<\/p>\n<p>(b) Is a bank account number a function of the balance?<\/p>\n<p>(c) Is a balance a one-to-one function of the bank account number?<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Evaluate the following:<\/p>\n<p>(a) If each percent grade earned in a course translates to one letter grade, is the letter grade a function of the percent grade?<\/p>\n<p>(b) If so, is the function one-to-one?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135435781\" data-depth=\"1\">\n<h2 data-type=\"title\">Using the Vertical Line Test<\/h2>\n<p id=\"fs-id1165135435786\">As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.<\/p>\n<p id=\"fs-id1165137637786\">The most common graphs name the input value\u00a0[latex]x[\/latex] and the output value [latex]y,[\/latex] and we say [latex]y[\/latex] is a function of [latex]x,[\/latex] or [latex]y=f(x)[\/latex] when the function is named [latex]f.[\/latex] The graph of the function is the set of all points [latex](x, y)[\/latex] in the plane that satisfies the equation [latex]y=f(x).[\/latex] If the function is defined for only a few input values, then the graph of the function is only a few points, where the <em data-effect=\"italics\">x<\/em>-coordinate of each point is an input value and the <em data-effect=\"italics\">y<\/em>-coordinate of each point is the corresponding output value. For example, the black dots on the graph in Figure 7 tell us that [latex]f(0)=2[\/latex] and [latex]f(6)=1.[\/latex] However, the set of all points [latex](x, y)[\/latex] satisfying [latex]y=f(x)[\/latex] is a curve. The curve shown includes [latex](0, 2)[\/latex] and [latex](6, 1)[\/latex] because the curve passes through those points.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_462\" aria-describedby=\"caption-attachment-462\" style=\"width: 604px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-462\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-7-300x220.jpg\" alt=\"\" width=\"604\" height=\"443\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-7-300x220.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-7-65x48.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-7-225x165.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-7-350x256.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-7.jpg 434w\" sizes=\"auto, (max-width: 604px) 100vw, 604px\" \/><figcaption id=\"caption-attachment-462\" class=\"wp-caption-text\">Figure 7<\/figcaption><\/figure>\n<p id=\"fs-id1165137737620\">The <strong><span id=\"term-00020\" data-type=\"term\">vertical line test<\/span><\/strong> can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does <em data-effect=\"italics\">not<\/em> define a function because a function has only one output value for each input value. See Figure 8.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_463\" aria-describedby=\"caption-attachment-463\" style=\"width: 708px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-463\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-8-300x103.jpg\" alt=\"\" width=\"708\" height=\"243\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-8-300x103.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-8-1024x352.jpg 1024w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-8-768x264.jpg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-8-65x22.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-8-225x77.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-8-350x120.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-8.jpg 1105w\" sizes=\"auto, (max-width: 708px) 100vw, 708px\" \/><figcaption id=\"caption-attachment-463\" class=\"wp-caption-text\">Figure 8<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a graph, use the vertical line test to determine if the graph represents a function.<\/strong><\/p>\n<ol>\n<li>Inspect the graph to see if any vertical line drawn would intersect the curve more than once.<\/li>\n<li>Is there is any such line, determine that the graph does not represent a function.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 14: Applying the Vertical Line Test<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Which of the graphs in Figure 9 represent(s) a function [latex]y=f(x)?[\/latex]<\/p>\n<figure id=\"attachment_464\" aria-describedby=\"caption-attachment-464\" style=\"width: 561px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-464\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-9-300x122.jpg\" alt=\"\" width=\"561\" height=\"228\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-9-300x122.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-9-768x313.jpg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-9-65x27.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-9-225x92.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-9-350x143.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-9.jpg 941w\" sizes=\"auto, (max-width: 561px) 100vw, 561px\" \/><figcaption id=\"caption-attachment-464\" class=\"wp-caption-text\">Figure 9<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>If any vertical line intersects the graph more than once, the relation represented by the graph is not a function. Notice that any vertical line would pass through only one point of the two graphs shown in parts (a) and (b) of Figure 9. From this we can conclude that these two graphs represent functions. The third graph does not represent a function because, as most x-values, a vertical line would intersect the graph at more than one point, as shown in Figure 10.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_465\" aria-describedby=\"caption-attachment-465\" style=\"width: 549px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-465\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-10-300x274.jpg\" alt=\"\" width=\"549\" height=\"501\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-10-300x274.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-10-65x59.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-10-225x206.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-10-350x320.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-10.jpg 487w\" sizes=\"auto, (max-width: 549px) 100vw, 549px\" \/><figcaption id=\"caption-attachment-465\" class=\"wp-caption-text\">Figure 10<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #11<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Does the graph in Figure 11 represent a function?<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_466\" aria-describedby=\"caption-attachment-466\" style=\"width: 515px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-466\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-11-287x300.jpg\" alt=\"\" width=\"515\" height=\"538\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-11-287x300.jpg 287w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-11-65x68.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-11-225x235.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-11.jpg 289w\" sizes=\"auto, (max-width: 515px) 100vw, 515px\" \/><figcaption id=\"caption-attachment-466\" class=\"wp-caption-text\">Figure 11<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137610952\" data-depth=\"1\">\n<h2 data-type=\"title\">Using the Horizontal Line Test<\/h2>\n<p id=\"fs-id1165137871503\">Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the <strong><span id=\"term-00021\" data-type=\"term\">horizontal line test<\/span><\/strong>. Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.<\/strong><\/p>\n<ul>\n<li>Inspect the graph to see if a horizontal line drawn would intersect the curve more than once,<\/li>\n<li>Is there is any such line, determine that the function is not one-to-one.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 15: Applying the Horizontal Line Test<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Consider the functions shown in Figure 9(a) and Figure 9(b). Are either of the functions one-to-one?<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The function in Figure 9(a) is not one-to-one. The horizontal line shown in Figure 12 intersects the graph of the function at two points (and we can even find horizontal lines that intersect it at three points).<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_468\" aria-describedby=\"caption-attachment-468\" style=\"width: 594px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-468\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-12-290x300.jpg\" alt=\"\" width=\"594\" height=\"614\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-12-290x300.jpg 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-12-65x67.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-12-225x233.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-fig-12.jpg 314w\" sizes=\"auto, (max-width: 594px) 100vw, 594px\" \/><figcaption id=\"caption-attachment-468\" class=\"wp-caption-text\">Figure 12<\/figcaption><\/figure>\n<p>The function in Figure 9(b) is one-to-one. Any horizontal line will intersect a diagonal line at most once.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #12<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Is the graph shown in Figure 9(c) one-to-one?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135545919\" data-depth=\"1\">\n<h2 data-type=\"title\">Identifying Basic Toolkit Functions<\/h2>\n<p id=\"fs-id1165137698132\">In this text, we will be exploring functions\u2014the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our \u201ctoolkit functions,\u201d which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use\u00a0[latex]x[\/latex] as the input variable and [latex]y=f(x)[\/latex] as the output variable.<\/p>\n<p id=\"fs-id1165135591070\">We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown in Table 13.<\/p>\n<div id=\"Table_01_01_14\" class=\"os-table\">\n<table class=\"grid\" data-id=\"Table_01_01_14\">\n<caption>Table 13<\/caption>\n<colgroup>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/><\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: center;\" colspan=\"3\" scope=\"colgroup\">Toolkit Functions<\/th>\n<\/tr>\n<tr>\n<th style=\"text-align: center;\" scope=\"col\" data-align=\"center\">Name<\/th>\n<th style=\"text-align: center;\" scope=\"col\" data-align=\"center\">Function<\/th>\n<th style=\"text-align: center;\" scope=\"col\" data-align=\"center\">Graph<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td style=\"text-align: center;\" data-align=\"center\">Constant<\/td>\n<td style=\"text-align: center;\" data-align=\"center\">[latex]f(x)-c,[\/latex] where [latex]c[\/latex] is a constant<\/td>\n<td style=\"text-align: center;\" data-align=\"center\"><span id=\"fs-id1165137643159\" data-type=\"media\" data-alt=\"Graph of a constant function.\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-469\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-constant-300x179.jpg\" alt=\"\" width=\"300\" height=\"179\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-constant-300x179.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-constant-65x39.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-constant-225x134.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-constant-350x209.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-constant.jpg 461w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\" data-align=\"center\">Identity<\/td>\n<td style=\"text-align: center;\" data-align=\"center\">[latex]f(x)=x[\/latex]<\/td>\n<td style=\"text-align: center;\" data-align=\"center\"><span id=\"fs-id1165137811013\" data-type=\"media\" data-alt=\"Graph of a straight line.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-470\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-identity-300x179.jpg\" alt=\"\" width=\"300\" height=\"179\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-identity-300x179.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-identity-65x39.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-identity-225x134.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-identity-350x209.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-identity.jpg 461w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\" data-align=\"center\">Absolute value<\/td>\n<td style=\"text-align: center;\" data-align=\"center\">[latex]f(x)=|x|[\/latex]<\/td>\n<td style=\"text-align: center;\" data-align=\"center\"><span id=\"fs-id1165135195221\" data-type=\"media\" data-alt=\"Graph of absolute function.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-471\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-absolute-300x179.jpg\" alt=\"\" width=\"300\" height=\"179\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-absolute-300x179.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-absolute-65x39.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-absolute-225x134.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-absolute-350x209.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-absolute.jpg 461w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\" data-align=\"center\">Quadratic<\/td>\n<td style=\"text-align: center;\" data-align=\"center\">[latex]f(x)=x^2[\/latex]<\/td>\n<td style=\"text-align: center;\" data-align=\"center\"><span id=\"fs-id1165137501903\" data-type=\"media\" data-alt=\"Graph of a parabola.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-472\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-quadratic-300x179.jpg\" alt=\"\" width=\"300\" height=\"179\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-quadratic-300x179.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-quadratic-65x39.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-quadratic-225x134.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-quadratic-350x209.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-quadratic.jpg 461w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\" data-align=\"center\">Cubic<\/td>\n<td style=\"text-align: center;\" data-align=\"center\">[latex]f(x)=x^3[\/latex]<\/td>\n<td style=\"text-align: center;\" data-align=\"center\"><span id=\"fs-id1165137722123\" data-type=\"media\" data-alt=\"Graph of f(x) = x^3.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-473\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-cubic-300x179.jpg\" alt=\"\" width=\"300\" height=\"179\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-cubic-300x179.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-cubic-65x39.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-cubic-225x134.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-cubic-350x209.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-cubic.jpg 461w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\" data-align=\"center\">Reciprocal<\/td>\n<td style=\"text-align: center;\" data-align=\"center\">[latex]f(x)=\\frac{1}{x}[\/latex]<\/td>\n<td style=\"text-align: center;\" data-align=\"center\"><span id=\"fs-id1165134544980\" data-type=\"media\" data-alt=\"Graph of f(x)=1\/x.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-474\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-reciprocal-300x179.jpg\" alt=\"\" width=\"300\" height=\"179\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-reciprocal-300x179.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-reciprocal-65x39.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-reciprocal-225x134.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-reciprocal-350x209.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-reciprocal.jpg 461w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\" data-align=\"center\">Reciprocal squared<\/td>\n<td style=\"text-align: center;\" data-align=\"center\">[latex]f(x)=\\frac{1}{x^2}[\/latex]<\/td>\n<td style=\"text-align: center;\" data-align=\"center\"><span id=\"fs-id1165137647610\" data-type=\"media\" data-alt=\"Graph of f(x)=1\/x^2.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-475\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-reciprocal-square-300x179.jpg\" alt=\"\" width=\"300\" height=\"179\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-reciprocal-square-300x179.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-reciprocal-square-65x39.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-reciprocal-square-225x134.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-reciprocal-square-350x209.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-reciprocal-square.jpg 461w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\" data-align=\"center\">Square root<\/td>\n<td style=\"text-align: center;\" data-align=\"center\">[latex]f(x)=\\sqrt{x}[\/latex]<\/td>\n<td style=\"text-align: center;\" data-align=\"center\"><span id=\"fs-id1165137863670\" data-type=\"media\" data-alt=\"Graph of f(x)=sqrt(x).\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-476\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-square-300x179.jpg\" alt=\"\" width=\"300\" height=\"179\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-square-300x179.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-square-65x39.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-square-225x134.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-square-350x209.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-square.jpg 461w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"text-align: center;\" data-align=\"center\">Cube root<\/td>\n<td style=\"text-align: center;\" data-align=\"center\">[latex]f(x)=\\sqrt[3]{x}[\/latex]<\/td>\n<td style=\"text-align: center;\" data-align=\"center\"><span id=\"fs-id1165137838612\" data-type=\"media\" data-alt=\"Graph of f(x)=x^(1\/3).\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-477\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-cube-300x179.jpg\" alt=\"\" width=\"300\" height=\"179\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-cube-300x179.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-cube-65x39.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-cube-225x134.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-cube-350x209.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1-cube.jpg 461w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><\/div>\n<div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Media<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Access the following online resources for additional instruction and practice with functions.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=zT69oxcMhPw\/\">Determine if a Relation is a Function<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=gO5WN9g1fJo\">Vertical Line Test<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=sW9-zBeQpCU\">Introduction to Functions<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=5Z8DaZPJLKY\/\">Vertical Line Test on Graph<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=QFOJmevha_Y&amp;feature=youtu.be\/\">One-to-one Functions<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=tbSGdcSN8RE&amp;feature=youtu.be\/\">Graphs as One-to-one Functions<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">3.1 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1165137737761\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1165137432988\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1165137432993\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137432995\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137432993-solution\">1<\/a><span class=\"os-divider\">. <\/span>What is the difference between a relation and a function?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137870912\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137870914\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>What is the difference between the input and the output of a function?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137870922\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134118508\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137870922-solution\">3<\/a><span class=\"os-divider\">. <\/span>Why does the vertical line test tell us whether the graph of a relation represents a function?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135570273\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135570275\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>How can you determine if a relation is a one-to-one function?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134391600\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134391602\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134391600-solution\">5<\/a><span class=\"os-divider\">. <\/span>Why does the horizontal line test tell us whether the graph of a function is one-to-one?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165134080937\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id1165134080942\">For the following exercises, determine whether the relation represents a function.<\/p>\n<div id=\"fs-id1165134080945\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134080947\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex]\\{(a, b), (c, d), (a, c)\\}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135570225\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135570227\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135570225-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex]\\{(a, b), (b, c), (c, c)\\}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135318984\">For the following exercises, determine whether the relation represents [latex]y[\/latex] as a function of [latex]x.[\/latex]<\/p>\n<div id=\"fs-id1165137841682\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137841684\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex]5x+2y=10[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137925514\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135628498\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137925514-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex]y=x^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137942483\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137722399\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex]x=y^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135394228\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135394230\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135394228-solution\">11<\/a><span class=\"os-divider\">.<\/span> [latex]3x^2+y=14[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135675195\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135173422\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex]2x+y^2=6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137864148\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134085981\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137864148-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex]y=-2x^2+40x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137679356\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137924374\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex]y=\\frac{1}{x}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137661060\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137661062\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137661060-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex]x=\\frac{3y+5}{7y-1}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135581214\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135581216\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex]x=\\sqrt{1-y^2}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133202429\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133202431\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165133202429-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex]y=\\frac{3x+5}{7x-1}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134042772\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134042774\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex]x^2+y^2=9[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137887434\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137887436\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137887434-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex]2xy=1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137658561\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137658563\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex]x=y^3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137777686\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137777688\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137777686-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex]y=x^3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137892513\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137892515\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex]y=\\sqrt{1-x^2}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134216874\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134216877\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134216874-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex]x=\\pm\\sqrt{1-y}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135538770\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135538772\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex]y=\\pm\\sqrt{1-x}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135203596\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135203598\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135203596-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex]y^2=x^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137599984\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137599986\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex]y^3=x^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165134066606\">For the following exercises, evaluate [latex]f(-3, f(2), f(-a), -f(a), f(a+h)[\/latex]<\/p>\n<div id=\"fs-id1165137431335\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137431337\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137431335-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=2x-5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137727203\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137603598\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=-5x^2+2x-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137844088\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137844090\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137844088-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\sqrt{2-x}+5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135697840\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135697842\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{6x-1}{5x+2}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135453854\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135453856\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135453854-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=|x-1|+|x+1|[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135195666\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135195668\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span>Given the function [latex]g(x)=5-x^2,[\/latex] evaluate [latex]\\frac{g(x+h)-g(x)}{h}, h\\not=0.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135579705\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135579707\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135579705-solution\">33<\/a><span class=\"os-divider\">. <\/span>Given the function [latex]g(x)=x^2+2x,[\/latex] evaluate [latex]\\frac{g(x)-g(a)}{x-a}, x\\not=a.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134036847\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134036849\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span>Given the function [latex]k(t)=2t-1:[\/latex]<\/p>\n<p>(a) Evaluate [latex]k(2).[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p>(b) Solve [latex]k(t)=7.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134155170\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134155172\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134155170-solution\">35<\/a><span class=\"os-divider\">. <\/span>Given the function [latex]f(x)=8-3x:[\/latex]<\/p>\n<p>(a) Evaluate [latex]f(-2).[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p>(b) Solve [latex]f(x)=-1.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137935719\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135388490\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span>Given the function [latex]- p(c)=c^2+c:[\/latex]<\/p>\n<p>(a) Evaluate [latex]p(-3).[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p>(b) Solve [latex]p(c)=2.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135361357\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135361359\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135361357-solution\">37<\/a><span class=\"os-divider\">. <\/span>Given the function [latex]f(x)=x^2-3x:[\/latex]<\/p>\n<p>(a) Evaluate [latex]f(5).[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p>(b) Solve [latex]f(x)=4.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137833947\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137833949\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span>Given the function [latex]f(x)=\\sqrt{x+2}:[\/latex]<\/p>\n<p>(a) Evaluate [latex]f(7).[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p>(b) Solve [latex]f(x)=4.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137433542\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137433544\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137433542-solution\">39<\/a><span class=\"os-divider\">. <\/span>Consider the relationship [latex]3r+2t=18.[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p>(a) Write the relationship as a function [latex]r=f(t).[\/latex]<\/p>\n<p>(b) Evaluate [latex]f(-3).[\/latex]<\/p>\n<p>(c) Solve [latex]f(t)=2.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135664071\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphical<\/h3>\n<p id=\"fs-id1165135664077\">For the following exercises, use the vertical line test to determine which graphs show relations that are functions.<\/p>\n<div id=\"fs-id1165135455987\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135455989\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1153\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.40-288x300.webp\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.40-288x300.webp 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.40-65x68.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.40-225x234.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.40-350x365.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.40.webp 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137527641\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137847086\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137527641-solution\">41<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165137847091\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1154\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.41-288x300.webp\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.41-288x300.webp 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.41-65x68.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.41-225x234.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.41-350x365.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.41.webp 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135332512\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133336399\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165133336405\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1155\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.42-288x300.webp\" alt=\"\" width=\"288\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.42-288x300.webp 288w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.42-65x68.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.42-225x234.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.42-350x365.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.42.webp 357w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137742393\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137742395\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137742393-solution\">43<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165137597394\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1156\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.43-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.43-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.43-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.43-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.43-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.43.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135386379\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135386381\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165135386387\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1157\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.44-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.44-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.44-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.44-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.44-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.44.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137749974\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137439464\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137749974-solution\">45<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165137439470\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1158\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.45-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.45-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.45-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.45-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.45-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.45.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137399704\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137399706\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165135704896\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1159\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.46-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.46-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.46-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.46-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.46-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.46.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137883764\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137883767\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137883764-solution\">47<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165137883773\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1160\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.47-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.47-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.47-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.47-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.47-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.47.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134497159\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134497161\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165134497168\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1161\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.48-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.48-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.48-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.48-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.48-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.48.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135496435\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135496437\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135496435-solution\">49<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165134234204\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1162\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.49-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.49-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.49-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.49-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.49-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.49.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137911653\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137911656\" data-type=\"problem\">\n<p><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165137786191\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1163\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.50-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.50-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.50-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.50-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.50-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.50.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135593325\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135593327\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135593325-solution\">51<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165135593333\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1165\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.51-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.51-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.51-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.51-225x233.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.51-350x363.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.51.webp 361w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134240968\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134054028\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1165134054030\">Given the following graph,<\/p>\n<p>(a) Evaluate [latex]f(-1).[\/latex]<\/p>\n<p>(b) Solve for [latex]f(x)=3.[\/latex]<\/p>\n<p><span id=\"fs-id1165137834413\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1166\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.52-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.52-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.52-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.52-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.52-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.52.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135632092\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135632095\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135632092-solution\">53<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1165137861992\">Given the following graph,<\/p>\n<p>(a) Evaluate [latex]f(0).[\/latex]<\/p>\n<p>(b) Solve for [latex]f(x)=-3.[\/latex]<\/p>\n<p><span id=\"fs-id1165135567425\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1167\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.53-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.53-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.53-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.53-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.53-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.53.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134325868\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134325870\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1165134325872\">Given the following graph,<\/p>\n<p>(a) Evaluate [latex]f(4).[\/latex]<\/p>\n<p>(b) Solve for [latex]f(x)=1.[\/latex]<\/p>\n<p><span id=\"fs-id1165135575950\" data-type=\"media\" data-alt=\"Graph of relation.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1168\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.54-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.54-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.54-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.54-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.54-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.54.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135531627\">For the following exercises, determine if the given graph is a one-to-one function.<\/p>\n<div id=\"fs-id1165135541711\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135541713\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135541711-solution\">55<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165135541720\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a circle.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1169\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.55-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.55-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.55-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.55-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.55-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.55.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133085674\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134380351\" data-type=\"problem\">\n<p><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165134380356\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a parabola.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1170\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.56-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.56-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.56-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.56-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.56-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.56.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134037560\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134037562\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134037560-solution\">57<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165134037568\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a rotated cubic function.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1171\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.57-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.57-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.57-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.57-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.57-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.57.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134031248\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134031250\" data-type=\"problem\">\n<p><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165134031257\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of half of 1\/x.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1172\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.58-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.58-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.58-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.58-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.58-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.58.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134394579\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134394581\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134394579-solution\">59<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165135457089\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a one-to-one function.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1173\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.59-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.59-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.59-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.59-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.59-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.1.59.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135342204\" data-depth=\"2\">\n<h3 data-type=\"title\">Numeric<\/h3>\n<p id=\"fs-id1165133324912\">For the following exercises, determine whether the relation represents a function.<\/p>\n<div id=\"fs-id1165133324915\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133324917\" data-type=\"problem\">\n<p><span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span> [latex]\\{(-1, -1), (-2, -2), (-3, -3)\\}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135245507\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135245509\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135245507-solution\">61<\/a><span class=\"os-divider\">. <\/span> [latex]\\{(3, 4), (4, 5), (5, 6)\\}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135381342\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137724837\" data-type=\"problem\">\n<p><span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span> [latex]\\{(2, 5), (7, 11), (15, 8), (7, 9)\\}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165133260452\">For the following exercises, determine if the relation represented in table form represents [latex]y[\/latex] as a function of [latex]x.[\/latex]<\/p>\n<div id=\"fs-id1165137644802\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137644804\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137644802-solution\">63<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<div id=\"fs-id1165137644806\" class=\"os-table first-element\">\n<table class=\"grid\" data-id=\"fs-id1165137644806\" data-label=\"\">\n<colgroup>\n<col \/>\n<col data-width=\"50\" \/>\n<col data-width=\"50\" \/>\n<col data-width=\"50\" \/> <\/colgroup>\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]x[\/latex]<\/td>\n<td data-align=\"center\">5<\/td>\n<td data-align=\"center\">10<\/td>\n<td data-align=\"center\">15<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]y[\/latex]<\/td>\n<td data-align=\"center\">3<\/td>\n<td data-align=\"center\">8<\/td>\n<td data-align=\"center\">14<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137771740\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137771742\" data-type=\"problem\">\n<p><span class=\"os-number\">64<\/span><span class=\"os-divider\">.<\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<div id=\"fs-id1165137771744\" class=\"os-table first-element\">\n<table class=\"grid\" data-id=\"fs-id1165137771744\" data-label=\"\">\n<colgroup>\n<col \/>\n<col data-width=\"50\" \/>\n<col data-width=\"50\" \/>\n<col data-width=\"50\" \/><\/colgroup>\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]x[\/latex]<\/td>\n<td data-align=\"center\">5<\/td>\n<td data-align=\"center\">10<\/td>\n<td data-align=\"center\">15<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]y[\/latex]<\/td>\n<td data-align=\"center\">3<\/td>\n<td data-align=\"center\">8<\/td>\n<td data-align=\"center\">8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137758640\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137758643\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137758640-solution\">65<\/a><span class=\"os-divider\">.<\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<div id=\"fs-id1165137758645\" class=\"os-table first-element\">\n<table class=\"grid\" style=\"height: 156px;\" data-id=\"fs-id1165137758645\" data-label=\"\">\n<colgroup>\n<col \/>\n<col data-width=\"50\" \/>\n<col data-width=\"50\" \/>\n<col data-width=\"50\" \/><\/colgroup>\n<tbody>\n<tr style=\"height: 78px;\">\n<td style=\"height: 78px; width: 138.783px;\" data-align=\"center\">[latex]x[\/latex]<\/td>\n<td style=\"height: 78px; width: 135.517px;\" data-align=\"center\">5<\/td>\n<td style=\"height: 78px; width: 179.15px;\" data-align=\"center\">10<\/td>\n<td style=\"height: 78px; width: 182.183px;\" data-align=\"center\">10<\/td>\n<\/tr>\n<tr style=\"height: 78px;\">\n<td style=\"height: 78px; width: 138.783px;\" data-align=\"center\">[latex]y[\/latex]<\/td>\n<td style=\"height: 78px; width: 135.517px;\" data-align=\"center\">3<\/td>\n<td style=\"height: 78px; width: 179.15px;\" data-align=\"center\">8<\/td>\n<td style=\"height: 78px; width: 182.183px;\" data-align=\"center\">14<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135641701\">For the following exercises, use the function [latex]f[\/latex] represented in the table below.<\/p>\n<div id=\"eip-reftable\" class=\"os-table\">\n<table class=\"grid\" data-id=\"eip-reftable\">\n<caption>Table 14<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 92px;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 61px;\">0<\/td>\n<td style=\"width: 64px;\">1<\/td>\n<td style=\"width: 46px;\">2<\/td>\n<td style=\"width: 64px;\">3<\/td>\n<td style=\"width: 65px;\">4<\/td>\n<td style=\"width: 47px;\">5<\/td>\n<td style=\"width: 65px;\">6<\/td>\n<td style=\"width: 64px;\">7<\/td>\n<td style=\"width: 58px;\">8<\/td>\n<td style=\"width: 62px;\">9<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 92px;\">[latex]f(x)[\/latex]<\/td>\n<td style=\"width: 61px;\">74<\/td>\n<td style=\"width: 64px;\">28<\/td>\n<td style=\"width: 46px;\">1<\/td>\n<td style=\"width: 64px;\">53<\/td>\n<td style=\"width: 65px;\">56<\/td>\n<td style=\"width: 47px;\">3<\/td>\n<td style=\"width: 65px;\">36<\/td>\n<td style=\"width: 64px;\">45<\/td>\n<td style=\"width: 58px;\">14<\/td>\n<td style=\"width: 62px;\">47<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>Evaluate [latex]f(3).[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137453742\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137453744\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137453742-solution\">67<\/a><span class=\"os-divider\">. <\/span>Solve [latex]f(x)=1.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137757773\">For the following exercises, evaluate the function [latex]f[\/latex] at the values [latex]f(-2), f(-1), f(0), f(1)[\/latex] and [latex]f(2).[\/latex]<\/p>\n<div id=\"fs-id1165135581074\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135581076\" data-type=\"problem\">\n<p><span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=4-2x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137812524\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137812526\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137812524-solution\">69<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=8-3x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135445749\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135445751\" data-type=\"problem\">\n<p><span class=\"os-number\">70<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=8x^2-7x+3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137937596\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135181211\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137937596-solution\">71<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=3+\\sqrt{x+3}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134573828\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134573830\" data-type=\"problem\">\n<p><span class=\"os-number\">72<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{x-2}{x+3}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133248574\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133248576\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165133248574-solution\">73<\/a><span class=\"os-divider\">. <\/span> [latex]f(x0=3^x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135306461\">For the following exercises, evaluate the expressions, given functions [latex]f, g[\/latex] and [latex]h:[\/latex]<\/p>\n<p>[latex]f(x)=3x-2[\/latex]<br \/>\n[latex]g(x)=5-x^2[\/latex]<br \/>\n[latex]h(x)=-2x^2+3x-1[\/latex]<\/p>\n<div id=\"fs-id1165135575197\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135575199\" data-type=\"problem\">\n<p><span class=\"os-number\">74<\/span><span class=\"os-divider\">. <\/span> [latex]3f(1)-4g(-2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134086037\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134086039\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134086037-solution\">75<\/a><span class=\"os-divider\">. <\/span> [latex]f(\\frac{7}{3})-h(-2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165134373511\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<p id=\"fs-id1165135530586\">For the following exercises, graph [latex]y=x^2[\/latex] on the given domain. Determine the corresponding range. Show each graph.<\/p>\n<div id=\"fs-id1165135530591\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135530593\" data-type=\"problem\">\n<p><span class=\"os-number\">76<\/span><span class=\"os-divider\">. <\/span> [latex][-0.1, 0.1][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134087674\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134087676\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134087674-solution\">77<\/a><span class=\"os-divider\">. <\/span> [latex][-10, 10][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135695182\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135695184\" data-type=\"problem\">\n<p><span class=\"os-number\">78<\/span><span class=\"os-divider\">. <\/span> [latex][-100, 100][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135388446\">For the following exercises, graph [latex]y=x^3[\/latex] on the given domain. Determine the corresponding range. Show each graph.<\/p>\n<div id=\"fs-id1165134060421\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134060424\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134060421-solution\">79<\/a><span class=\"os-divider\">. <\/span> [latex][-0.1, 0.1][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133195224\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133195226\" data-type=\"problem\">\n<p><span class=\"os-number\">80<\/span><span class=\"os-divider\">. <\/span> [[latex][-10, 10][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133402070\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133402072\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165133402070-solution\">81<\/a><span class=\"os-divider\">. <\/span> [latex][-100, 100][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135634160\">For the following exercises, graph [latex]y=\\sqrt{x}[\/latex] on the given domain. Determine the corresponding range. Show each graph.<\/p>\n<div id=\"fs-id1165137844302\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137844304\" data-type=\"problem\">\n<p><span class=\"os-number\">82<\/span><span class=\"os-divider\">. <\/span> [latex][0, 0.01][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137540730\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137540733\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137540730-solution\">83<\/a><span class=\"os-divider\">. <\/span> [latex][0, 100][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137605840\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137605842\" data-type=\"problem\">\n<p><span class=\"os-number\">84<\/span><span class=\"os-divider\">. <\/span> [latex][0, 10,000][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137539207\">For the following exercises, graph [latex]y=\\sqrt[3]{x}[\/latex] on the given domain. Determine the corresponding range. Show each graph.<\/p>\n<div id=\"fs-id1165134031227\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134031229\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134031227-solution\">85<\/a><span class=\"os-divider\">. <\/span> [latex][-0.001, 0.001][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134087649\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134087651\" data-type=\"problem\">\n<p><span class=\"os-number\">86<\/span><span class=\"os-divider\">. <\/span> [latex][-1000, 1000][\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135251229\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135251230\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165135251229-solution\">87<\/a><span class=\"os-divider\">. <\/span> [latex][-1,000,000, 1,000,000[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135580349\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<div id=\"fs-id1165135580355\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135580357\" data-type=\"problem\">\n<p><span class=\"os-number\">88<\/span><span class=\"os-divider\">. <\/span>The amount of garbage, [latex]G[\/latex] produced by a city with population [latex]p[\/latex] is given by [latex]G=f(p).[\/latex] [latex]G[\/latex] is measured in tons per week, and [latex]p[\/latex] is measured in thousands of people.<\/p>\n<div class=\"os-problem-container\">\n<p>(a) The town of Windsor, Colorado, has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function [latex]f.[\/latex]<\/p>\n<p>(b) Explain the meaning of the statement [latex]f(5)=2.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137922382\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134269005\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165137922382-solution\">89<\/a><span class=\"os-divider\">. <\/span>The number of cubic yards of dirt, [latex]D,[\/latex] needed to cover a garden with area [latex]a[\/latex] square feet is given by [latex]D=g(a).[\/latex]<\/p>\n<div class=\"os-problem-container\">\n<p>(a) A garden with area [latex]5000\\hspace{0.5em}\\text{ft}^2[\/latex] requires [latex]50\\hspace{0.5em}\\text{yd}^3[\/latex] of dirt. Express this information in terms of the function [latex]g.[\/latex]<\/p>\n<p>(b) Explain the meaning of the statement [latex]g(100)=1.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135553615\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135553617\" data-type=\"problem\">\n<p><span class=\"os-number\">90<\/span><span class=\"os-divider\">. <\/span>Let [latex]f(t)[\/latex] be the number of ducks in a lake [latex]t[\/latex] years after 1990. Explain the meaning of each statement:<\/p>\n<div class=\"os-problem-container\">\n<p>(a) [latex]f(5)=30[\/latex]<\/p>\n<p>(b) [latex]f(10)=40[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134272734\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134272736\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-3\" data-page-slug=\"chapter-3\" data-page-uuid=\"9b8c8027-99b3-55ba-86c1-ba45e9ff3eff\" data-page-fragment=\"fs-id1165134272734-solution\">91<\/a><span class=\"os-divider\">. <\/span>Let [latex]h(t)[\/latex] be the height above ground, in feet, of a rocket [latex]t[\/latex] seconds after launching. Explain the meaning of each statement:<\/p>\n<div class=\"os-problem-container\">\n<p>(a) [latex]h(1)=200[\/latex]<\/p>\n<p>(b) [latex]h(2)=350[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135708043\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135708045\" data-type=\"problem\">\n<p><span class=\"os-number\">92<\/span><span class=\"os-divider\">. <\/span>Show that the function [latex]f(x)=3(x-5)^2+7[\/latex] is <u>not<\/u>\u00a0one-to-one.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-132-1\">http:\/\/www.kgbanswers.com\/how-long-is-a-dogs-memory-span\/4221590. Accessed 3\/24\/2014. <a href=\"#return-footnote-132-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":158,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-132","chapter","type-chapter","status-publish","hentry"],"part":105,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/132","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/users\/158"}],"version-history":[{"count":24,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/132\/revisions"}],"predecessor-version":[{"id":1486,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/132\/revisions\/1486"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/105"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/132\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=132"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=132"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=132"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=132"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}