{"id":133,"date":"2025-04-09T17:15:05","date_gmt":"2025-04-09T17:15:05","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/3-2-domain-and-range-college-algebra-2e-openstax\/"},"modified":"2025-08-19T17:00:51","modified_gmt":"2025-08-19T17:00:51","slug":"3-2-domain-and-range","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/3-2-domain-and-range\/","title":{"raw":"3.2 Domain and Range","rendered":"3.2 Domain and Range"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_562c3737-a93d-458c-98c0-a04f442f13bd\" 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>Find the domain of a function defined by an equation.<\/li>\r\n \t<li>Graph piecewise-defined functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<section>\r\n<ul id=\"list-00001\"><\/ul>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137404978\">Horror and thriller movies are both popular and, very often, extremely profitable. When big-budget actors, shooting locations, and special effects are included, however, studios count on even more viewership to be successful. Consider five major thriller\/horror entries from the early 2000s\u2014<em data-effect=\"italics\">I am Legend<\/em>, <em data-effect=\"italics\">Hannibal<\/em>, <em data-effect=\"italics\">The Ring<\/em>, <em data-effect=\"italics\">The Grudge<\/em>, and <em data-effect=\"italics\">The Conjuring<\/em>. Figure 1 shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the <span id=\"term-00007\" class=\"no-emphasis\" data-type=\"term\">domain<\/span> and range. In this section, we will investigate methods for determining the domain and range of functions such as these.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_480\" align=\"aligncenter\" width=\"539\"]<img class=\"wp-image-480\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-1-300x114.jpg\" alt=\"\" width=\"539\" height=\"205\" \/> Figure 1.[\/caption]\r\n\r\n<section id=\"fs-id1165135193832\" data-depth=\"1\">\r\n<p data-type=\"title\"><span class=\"os-caption\">(Based on data compiled by www.the-numbers.com.[footnote]The Numbers: Where Data and the Movie Business Meet. \u201cBox Office History for Horror Movies.\u201d http:\/\/www.the-numbers.com\/market\/genre\/Horror. Accessed 3\/24\/2014[\/footnote])<\/span><\/p>\r\n\r\n<h2 data-type=\"title\">Finding the Domain of a Function Defined by an Equation<\/h2>\r\n<p id=\"fs-id1165135445896\">In Functions and Function Notation, we were introduced to the concepts of <span id=\"term-00008\" class=\"no-emphasis\" data-type=\"term\">domain and range<\/span>. In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.<\/p>\r\n<p id=\"fs-id1165135453892\">We can visualize the domain as a \u201cholding area\u201d that contains \u201craw materials\u201d for a \u201cfunction machine\u201d and the range as another \u201cholding area\u201d for the machine\u2019s products. See Figure 2.<\/p>\r\n&nbsp;\r\n\r\n<\/section>\r\n\r\n[caption id=\"attachment_481\" align=\"aligncenter\" width=\"670\"]<img class=\" wp-image-481\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-2-300x141.jpg\" alt=\"\" width=\"670\" height=\"315\" \/> Figure 2[\/caption]\r\n\r\n<section id=\"fs-id1165135193832\" data-depth=\"1\">\r\n<p id=\"fs-id1165137761714\">We can write the <span id=\"term-00009\" class=\"no-emphasis\" data-type=\"term\">domain and range<\/span> in <span id=\"term-00010\" data-type=\"term\">interval notation<\/span>, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, they would need to express the interval that is more than 0 and less than or equal to 100 and write [latex] (0, 100]. [\/latex] We will discuss interval notation in greater detail later.<\/p>\r\n<p id=\"fs-id1165135320406\">Let\u2019s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an odd root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function\u2019s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.<\/p>\r\n<p id=\"fs-id1165137552233\">Before we begin, let us review the conventions of interval notation:<\/p>\r\n\r\n<ul id=\"fs-id1165135673417\">\r\n \t<li>The smallest number from the interval is written first.<\/li>\r\n \t<li>The largest number in the interval is written second, following a comma.<\/li>\r\n \t<li>Parentheses, ( or ), are used to signify that an endpoint value is not included, called exclusive.<\/li>\r\n \t<li>Brackets, [ or ], are used to indicate that an endpoint value is included, called inclusive.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137807384\">See Figure 3 for a summary of interval notation.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_482\" align=\"aligncenter\" width=\"588\"]<img class=\" wp-image-482\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-3-300x278.jpg\" alt=\"\" width=\"588\" height=\"545\" \/> Figure 3[\/caption]\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 1: Finding the Domain of a Function as a Set of Ordered Pairs<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the <span id=\"term-00011\" class=\"no-emphasis\" data-type=\"term\">domain<\/span> of the following function: [latex] \\{(2, 10), (3, 10), (4, 20), (5, 30), (6, 40)\\} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>First identify the input values. The input value is the first coordinate in an ordered pair. There are no restrictions, as the ordered pairs are simple listed. The domain is the best of the first coordinates of the ordered pairs.\r\n<p style=\"text-align: center;\">[latex] \\{2, 3, 4, 5, 6\\} [\/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\">Try It #1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the domain of the function: [latex] \\{(-5, 4), (0, 0), (5, -4), (10, -8), (15, -12)\\} [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/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 written in equation form, find the domain.<\/strong>\r\n<ol>\r\n \t<li>Identify the input values.<\/li>\r\n \t<li>Identify any restrictions on the input and exclude those values from the domain.<\/li>\r\n \t<li>Write the domain in interval form, if possible.<\/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 2: Finding the Domain of a Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the domain of the function [latex] f(x)=x^2-1. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The input value, shown by the variable\u00a0[latex] x [\/latex] in the equation, is squared and then the result is lowered by one. Any real number may be squared and then lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers.\r\n\r\nIn interval dorm, the domain of\u00a0[latex] f [\/latex] is [latex] (-\\infty, \\infty). [\/latex]\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\nFind the domain of the function: [latex] f(x)=5-x+x^2. [\/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\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a function written in an equation form that includes a fraction, find the domain.<\/strong>\r\n<ol>\r\n \t<li>Identify the input values.<\/li>\r\n \t<li>Identify any restrictions on the input. If there is a denominator in the function's formula, set the denominator equal to zero and solve for x. If the function's formula contains an even root, set the radicand greater than or equal to 0, and then solve.<\/li>\r\n \t<li>Write the domain in interval dorm, making sure to exclude any restricted values from the domain.<\/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 3: Finding the Domain of a Function Involving a Denominator<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the domain of the function [latex] f(x)=\\frac{x+1}{2-x} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for [latex] x. [\/latex]\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{ll} 2-x &amp; =0 \\\\ -x &amp; =-2 \\\\ x &amp; =2 \\end{array} [\/latex]<\/p>\r\nNow, we will exclude 2 from the domain. The answers are all real numbers where [latex] x&lt;2 [\/latex] or [latex] x&gt;2 [\/latex] as shown in Figure 4. We can use a symbol known as the union, [latex] \\cup [\/latex] to combine the two sets. In interval notation, we write the solution: [latex] (-\\infty, 2)\\cup(2, \\infty) [\/latex]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_483\" align=\"aligncenter\" width=\"431\"]<img class=\" wp-image-483\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-4-300x101.jpg\" alt=\"\" width=\"431\" height=\"145\" \/> Figure 4[\/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 #3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the domain of the function: [latex] f(x)=\\frac{1+4x}{2x-1} [\/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\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a function written in equation form including an even root, find the domain.<\/strong>\r\n<ol>\r\n \t<li>Identify the input values.<\/li>\r\n \t<li>Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for x.<\/li>\r\n \t<li>The solution(s) are the domain of the function. If possible, write the answer in interval form.<\/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 4: Find the Domain of a Function with an Even Root<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the domain of the function [latex] f(x)=\\sqrt{7-x} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.\r\n\r\nSet the radicand greater than or equal to zero and solve for [latex] x. [\/latex]\r\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{ll} 7-x &amp; \\ge0 \\\\ -x &amp; \\ge-7 \\\\ x &amp; \\le7 \\end{array} [\/latex]<\/p>\r\nNow, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal to 7, or [latex] (-\\infty, 7]. [\/latex]\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\nFind the domain of the function [latex] f(x)=\\sqrt{5+2x}. [\/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: Can there be functions in which the domain and range do not intersect at all?<\/strong>\r\n\r\n<em>A: Yes. For example, the function\u00a0[latex] f(x)=-\\frac{1}{\\sqrt{x}} [\/latex] <\/em><em>has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a functions inputs and output can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.<\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165137677916\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using Notations to Specify Domain and Range<\/h2>\r\n<p id=\"fs-id1165137410091\">In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in <strong><span id=\"term-00015\" data-type=\"term\">set-builder notation<\/span><\/strong>. For example,\u00a0[latex] \\{x|10\\le x\\le30\\} [\/latex] describes the behavior of\u00a0[latex] x [\/latex] in set-builder notation. The braces\u00a0[latex] \\{\\} [\/latex] are read as \u201cthe set of,\u201d and the vertical bar | is read as \u201csuch that,\u201d so we would read\u00a0[latex] \\{x|10\\le x\\le30\\} [\/latex] as \u201cthe set of <em data-effect=\"italics\">x<\/em>-values such that 10 is less than or equal to\u00a0[latex] x [\/latex] and\u00a0[latex] x [\/latex] is less than 30.\u201d<\/p>\r\n<p id=\"fs-id1165135207589\">Figure 5 compares inequality notation, set-builder notation, and interval notation.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_484\" align=\"aligncenter\" width=\"600\"]<img class=\" wp-image-484\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-5-300x222.jpg\" alt=\"\" width=\"600\" height=\"444\" \/> Figure 5[\/caption]\r\n<p id=\"fs-id1165137911528\">To combine two intervals using inequality notation or set-builder notation, we use the word \u201cor.\u201d As we saw in earlier examples, we use the union symbol,\u00a0[latex] \\cup [\/latex]\u00a0to combine two unconnected intervals. For example, the union of the sets\u00a0[latex] \\{2, 3, 5\\} [\/latex] and\u00a0[latex] \\{4, 6\\} [\/latex] is the set\u00a0[latex] \\{2, 3, 4, 5, 6\\} [\/latex] It is the set of all elements that belong to one <em data-effect=\"italics\">or<\/em> the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is<\/p>\r\n\r\n<math display=\"block\"><\/math>\r\n<p style=\"text-align: center;\">[latex] \\{x| \\ \\ \\ |x| \\ge3\\}=(-\\infty, -3]\\cup[3, \\infty) [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Set-Builder Notation and Interval Notation<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Set-builder notation<\/strong> is a method of specifying a set of elements that satisfy a certain condition. It takes the form\u00a0[latex] \\{x|\\hspace{0.5em}\\text{statement about}\\hspace{0.25em}x\\} [\/latex] which is read as, \"the set of all x such that the statement about x is true. For example,\r\n<p style=\"text-align: center;\">[latex] \\{x|4&lt; x\\le12\\} [\/latex]<\/p>\r\n<p style=\"text-align: left;\"><strong>Interval notation<\/strong> is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. For example,<\/p>\r\n<p style=\"text-align: center;\">[latex] (4, 12] [\/latex]<\/p>\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\n<strong>Given a line graph, describe the set of values using interval notation.<\/strong>\r\n<ol>\r\n \t<li>Identify the intervals to be included in the set by determining where the heavy line overlays the real line.<\/li>\r\n \t<li>At the left end of each interval, use [ with each end value to be included in the set (solid dot) or ( for each excluded end value (open dot).<\/li>\r\n \t<li>At the right end of each interval, use ] with each end value to be included in the set (filled dot) or ) for each excluded end value (open dot).<\/li>\r\n \t<li>Use the union symbol\u00a0[latex] \\cup [\/latex] to combine all intervals into one set.<\/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: Describing Sets on the Real-Number Line<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDescribe the intervals of values shown in Figure 6 using inequality notation, set-builder notation, and interval notation.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_485\" align=\"aligncenter\" width=\"385\"]<img class=\" wp-image-485\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-6-300x39.jpg\" alt=\"\" width=\"385\" height=\"50\" \/> Figure 6[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>To describe the values,\u00a0[latex] x, [\/latex] included in the intervals shown, we would say, \"[latex] x [\/latex] is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.\"\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 45px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 50%; height: 15px; text-align: center;\"><strong>Inequality<\/strong><\/td>\r\n<td style=\"width: 50%; height: 15px; text-align: center;\">[latex] 1\\le x \\le3\u00a0 \\ \\text{or} \\ x&gt;5 [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 50%; height: 15px; text-align: center;\"><strong>Set-builder notation<\/strong><\/td>\r\n<td style=\"width: 50%; height: 15px; text-align: center;\">[latex] \\{x|1 \\le x\\le3 \\ \\text{or} \\ x&gt;5\\} [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 50%; height: 15px; text-align: center;\"><strong>Interval notation<\/strong><\/td>\r\n<td style=\"width: 50%; height: 15px; text-align: center;\">[latex] [1, 3]\\cup(5, \\infty) [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nRemember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. Using a parenthesis means the boundary is not included in the set.\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 Figure 7, specify the graphed set in\r\n\r\n(a) words\r\n\r\n(b) set-builder notation\r\n\r\n(c) interval notation\r\n\r\n[caption id=\"attachment_486\" align=\"aligncenter\" width=\"393\"]<img class=\" wp-image-486\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-7-300x45.jpg\" alt=\"\" width=\"393\" height=\"59\" \/> Figure 7[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137653855\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Finding Domain and Range from Graphs<\/h2>\r\n<p id=\"fs-id1165135161404\">Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the <em data-effect=\"italics\">x<\/em>-axis. The range is the set of possible output values, which are shown on the <em data-effect=\"italics\">y<\/em>-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See Figure 8.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_487\" align=\"aligncenter\" width=\"555\"]<img class=\" wp-image-487\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-8-230x300.jpg\" alt=\"\" width=\"555\" height=\"724\" \/> Figure 8[\/caption]\r\n<p id=\"fs-id1165137597994\">We can observe that the graph extends horizontally from\u00a0[latex] -5 [\/latex] to the right without bound, so the domain is\u00a0[latex] [-5, \\infty). [\/latex] The vertical extent of the graph is all range values\u00a0[latex] 5 [\/latex] and below, so the range is\u00a0[latex] (-\\infty, 5]. [\/latex] Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 6: Finding Domain and Range from a Graph<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the domain and range of the function f whose graph is shown in Figure 9.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_488\" align=\"aligncenter\" width=\"358\"]<img class=\" wp-image-488\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-9-300x208.jpg\" alt=\"\" width=\"358\" height=\"248\" \/> Figure 9[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We can observe that the horizontal extent of the graph is -3 to 1, so the domain of\u00a0[latex] f [\/latex]\u00a0is [latex] (-3, 1]. [\/latex]\r\n\r\nThe vertical extent of the graph is 0 to -4, so the range is\u00a0[latex] [-4, 0]. [\/latex] See Figure 10.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_489\" align=\"aligncenter\" width=\"508\"]<img class=\" wp-image-489\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-10-300x224.jpg\" alt=\"\" width=\"508\" height=\"379\" \/> 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\">Example 7: Finding Domain and Range from a Graph of iPhone Sales<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the domain and range of the function [latex] f [\/latex] whose graph is shown in Figure 11.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_492\" align=\"aligncenter\" width=\"550\"]<img class=\" wp-image-492\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-300x211.png\" alt=\"\" width=\"550\" height=\"387\" \/> Figure 11[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The input quantity along the horizontal axis is \"years\" which we represent with the variable [latex] t [\/latex] for time. The output quantity is \"billions of dollars per year,\" which we represent with the variable [latex] d [\/latex] for dollars. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as [latex] 2019\\le t\\le2023 [\/latex] and the range as approximately\u00a0[latex] 137.7\\le d\\le205.5 [\/latex]\r\n\r\nIn interval notation, the domain is\u00a0[latex] [2019, 2023] [\/latex] and the range is\u00a0[latex] [137.7, 205.5]. [\/latex] For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.\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 #6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven Figure 12, identify the domain and range using interval notation.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_493\" align=\"aligncenter\" width=\"468\"]<img class=\" wp-image-493\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-12-300x204.jpg\" alt=\"\" width=\"468\" height=\"318\" \/> Figure 12[\/caption]\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: Can a function's domain and range be the same?<\/strong>\r\n\r\n<em>A: Yes. For example, the domain and range of the cube root function are both the set of all real numbers.<\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165134384565\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Finding Domains and Ranges of the Toolkit Functions<\/h2>\r\n<p id=\"fs-id1165137419914\">We will now return to our set of toolkit functions to determine the domain and range of each.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_494\" align=\"aligncenter\" width=\"370\"]<img class=\" wp-image-494\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-13-259x300.jpg\" alt=\"\" width=\"370\" height=\"429\" \/> Figure 13. For the <strong>constant function<\/strong> [latex] f(x)=c, [\/latex] the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant [latex] c, [\/latex] so the range is the set [latex] \\{c\\} [\/latex] that contains this single element. In interval notation, this is written as [latex] [c, c], [\/latex] the interval that both begins and ends with [latex] c. [\/latex][\/caption][caption id=\"attachment_495\" align=\"aligncenter\" width=\"273\"]<img class=\" wp-image-495\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-14-264x300.jpg\" alt=\"\" width=\"273\" height=\"310\" \/> Figure 14. For the <strong>identity function<\/strong> [latex] f(x)=x, [\/latex] there is no restriction on [latex] x. [\/latex] Both the domain and range are the set of all real numbers.[\/caption]\u00a0[caption id=\"attachment_496\" align=\"aligncenter\" width=\"288\"]<img class=\" wp-image-496\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-15-262x300.jpg\" alt=\"\" width=\"288\" height=\"330\" \/> Figure 15. For the <strong>absolute value function<\/strong> [latex] f(x)=|x|, [\/latex] there is no restriction on [latex] x. [\/latex] However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0.[\/caption]\u00a0[caption id=\"attachment_497\" align=\"aligncenter\" width=\"318\"]<img class=\" wp-image-497\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-16-300x290.jpg\" alt=\"\" width=\"318\" height=\"307\" \/> Figure 16. For the <strong>quadratic function<\/strong> [latex] f(x)=x^2 [\/latex] the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.[\/caption]\u00a0[caption id=\"attachment_498\" align=\"aligncenter\" width=\"278\"]<img class=\" wp-image-498\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-17-262x300.jpg\" alt=\"\" width=\"278\" height=\"318\" \/> Figure 17. For the <strong>cubic function<\/strong> [latex] f(x)=x^3 [\/latex] the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.[\/caption]\u00a0[caption id=\"attachment_499\" align=\"aligncenter\" width=\"287\"]<img class=\" wp-image-499\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-18-259x300.jpg\" alt=\"\" width=\"287\" height=\"332\" \/> Figure 18. For the <strong>reciprocal function<\/strong> [latex] f(x)=\\frac{1}{x} [\/latex] we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also write [latex] \\{x|\\hspace{0.5em}x\\not=0\\} [\/latex] the set of all real numbers that are not zero.[\/caption]\u00a0[caption id=\"attachment_500\" align=\"aligncenter\" width=\"316\"]<img class=\" wp-image-500\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-19-300x300.jpg\" alt=\"\" width=\"316\" height=\"316\" \/> Figure 19. For the <strong>reciprocal squared function<\/strong> [latex] f(x)=\\frac{1}{x^2} [\/latex] we cannot divide by 0, so we must exclude 0 from the domain. There is also no [latex] x [\/latex] that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.[\/caption]\u00a0[caption id=\"attachment_501\" align=\"aligncenter\" width=\"310\"]<img class=\"wp-image-501\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-20-300x297.jpg\" alt=\"\" width=\"310\" height=\"307\" \/> Figure 20. For the <strong>square root function<\/strong> [latex] f(x)=\\sqrt{x}, [\/latex] we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number x is defined to be positive, even though the square of the negative number [latex] -\\sqrt{x} [\/latex] also gives us [latex] x. [\/latex][\/caption]<math display=\"inline\"><semantics><mrow><\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n[caption id=\"attachment_502\" align=\"aligncenter\" width=\"314\"]<img class=\" wp-image-502\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-21-300x292.jpg\" alt=\"\" width=\"314\" height=\"306\" \/> Figure 21. For the <strong>cube root function<\/strong> [latex] f(x)=\\sqrt[3]{x}, [\/latex] the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function).[\/caption]\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, determine the domain and range.<\/strong>\r\n<ol>\r\n \t<li>Exclude from the domain any input values that result in division by zero.<\/li>\r\n \t<li>Exclude from the domain any input values that have nonreal (or undefined) number outputs.<\/li>\r\n \t<li>Use the valid input values to determine the range of the output values.<\/li>\r\n \t<li>Look at the function graph and table values to confirm the actual function behavior.<\/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 8: Finding the Domain and Range Using Toolkit Functions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the domain and range of [latex] f(x)=2x^3-x. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.\r\n\r\nThe domain is\u00a0[latex] (-\\infty, \\infty) [\/latex] and the range is also [latex] (-\\infty, \\infty). [\/latex]\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 9: Finding the Domain and Range<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the domain and range of [latex] f(x)=\\frac{2}{x+1}. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We cannot evaluate the function at\u00a0[latex] -1 [\/latex] because division by zero is undefined. The domain is\u00a0[latex] (-\\infty, -1)\r\n\\cup(01, \\infty). [\/latex] Because the function is never zero, we exclude 0 from the range. The range is [latex] (-\\infty, 0)\\cup(0, \\infty). [\/latex]\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: Finding the Domain and Range<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the domain and range of [latex] f(x)=2\\sqrt{x+4}. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.\r\n<p style=\"text-align: center;\">[latex] x+r\\ge0 \\ \\text{when} \\ x\\ge-4 [\/latex]<\/p>\r\nThe domain of\u00a0[latex] f(x) [\/latex] is [latex] [-4, \\infty). [\/latex]\r\n\r\nWe then find the range. We know that\u00a0[latex] f(-4)=0, [\/latex] and the function value increases as [latex] x [\/latex] increases without any upper limit. We conclude that the range of\u00a0[latex] f [\/latex] is [latex] [0, \\infty). [\/latex]\r\n<h3>Analysis<\/h3>\r\nFigure 22 represents the function [latex] f. [\/latex]\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_503\" align=\"aligncenter\" width=\"318\"]<img class=\" wp-image-503\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-22-300x218.jpg\" alt=\"\" width=\"318\" height=\"231\" \/> Figure 22[\/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 #7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the domain and range of [latex] f(x)=-\\sqrt{2-x}. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135440477\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Graphing Piecewise-Defined Functions<\/h2>\r\n<p id=\"fs-id1165137409262\">Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function\u00a0[latex] f(x)=|x|. [\/latex] With a domain of all real numbers and a range of values greater than or equal to 0, <span id=\"term-00018\" class=\"no-emphasis\" data-type=\"term\">absolute value<\/span> can be defined as the <span id=\"term-00019\" class=\"no-emphasis\" data-type=\"term\">magnitude<\/span>, or <span id=\"term-00020\" class=\"no-emphasis\" data-type=\"term\">modulus<\/span>, of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.<\/p>\r\n<p id=\"fs-id1165137558775\">If we input 0, or a positive value, the output is the same as the input.<\/p>\r\n<p style=\"text-align: center;\">[latex] f(x)=x \\ \\text{if} \\ x\\ge0 [\/latex]<\/p>\r\n<p id=\"fs-id1165137529947\">If we input a negative value, the output is the opposite of the input.<\/p>\r\n<p style=\"text-align: center;\">[latex] f(x)=-x \\ \\text{if} \\ x&lt;0 [\/latex]<\/p>\r\n<p id=\"fs-id1165137863778\">Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A <strong><span id=\"term-00021\" data-type=\"term\">piecewise function<\/span><\/strong> is a function in which more than one formula is used to define the output over different pieces of the domain.<\/p>\r\n<p id=\"fs-id1165134042316\">We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain \u201cboundaries.\u201d For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income\u00a0[latex] S [\/latex] would be\u00a0[latex] 0.1S [\/latex] if\u00a0[latex] S\\le\\$10,000 [\/latex] and\u00a0[latex] \\$1000+0.2(S-\\$10,000) [\/latex] if [latex] S&gt;\\$10,000. [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Piecewise Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA <strong>piecewise function<\/strong> is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:\r\n<p style=\"text-align: center;\">[latex] f(x) = \\begin{cases}\\text{formula 1} &amp; \\text{if } x \\text{ is in domain 1} \\\\\\text{formula 2} &amp; \\text{if } x \\text{ is in domain 2} \\\\\\text{formula 3} &amp; \\text{if } x \\text{ is in domain 3}\\end{cases} [\/latex]<\/p>\r\nIn piecewise notation, the absolute value function is\r\n<p style=\"text-align: center;\">[latex] |x| = \\begin{cases}x \\quad \\text{ if } x \\geq 0 \\\\-x \\quad \\text{ if } x &lt; 0\\end{cases}[\/latex]<\/p>\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\n<strong>Given a piecewise function, write the formula and identify the domain for each interval.<\/strong>\r\n<ol>\r\n \t<li>Identify the intervals for which different rules apply.<\/li>\r\n \t<li>Determine formulas that describe how to calculate an output from an input in each interval.<\/li>\r\n \t<li>Use braces and if-statements to write the 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 11: Writing a Piecewise Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe Aurora History Museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a <span id=\"term-00022\" class=\"no-emphasis\" data-type=\"term\">function<\/span> relating the number of people,\u00a0[latex] n, [\/latex] to the cost, [latex] C. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Two different formulas will be needed. For [latex] n [\/latex]-values under 10,\u00a0[latex] C=5n. [\/latex] For values of [latex] n [\/latex] that are 10 or greater, [latex] C=50. [\/latex]\r\n<p style=\"text-align: center;\">[latex] C(n) = \\begin{cases}5n &amp; \\text{if} &amp; 0 &lt; n &lt; 10 \\\\50 &amp; \\text{if} &amp; n \\geq 10\\end{cases} [\/latex]<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nThe function is represented in Figure 23. The graph is a diagonal line from\u00a0[latex] n=0 [\/latex] to\u00a0[latex] n-10 [\/latex] and a constant after that. In this example, the two formulas agree at the meeting point where\u00a0[latex] n=10, [\/latex] but not all piecewise functions have this property.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_504\" align=\"aligncenter\" width=\"338\"]<img class=\" wp-image-504\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-23-300x245.jpg\" alt=\"\" width=\"338\" height=\"276\" \/> Figure 23[\/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\">Example 12: Working with a Piecewise Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA cell phone company uses the function below to determine the cost,\u00a0[latex] C, [\/latex] in dollars for\u00a0[latex] g [\/latex] gigabytes of data transfer.\r\n<p style=\"text-align: center;\">[latex] C(g) = \\begin{cases}25 &amp; \\text{if} &amp; 0 &lt; g &lt; 2 \\\\25 + 10(g - 2) &amp; \\text{if} &amp; g \\geq 2\\end{cases} [\/latex]<\/p>\r\nFind the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>To find the cost of using 1.5 gigabytes of data,\u00a0[latex] C(1.5), [\/latex] we first look to see which part of the domain our input falls in. Because 1.5 is less than 2, we use the first formula.\r\n<p style=\"text-align: center;\">[latex] C(1.5)=\\$25 [\/latex]<\/p>\r\nTo find the cost of using 4 gigabytes of data,\u00a0[latex] C(4), [\/latex] we see that our input of 4 is greater than 2, so we use the second formula.\r\n<p style=\"text-align: center;\">[latex] C(4)=25+10(4-2)=\\$45 [\/latex]<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nThe function is represented in Figure 24. We can see where the function changes from a constant to a shifted and stretched identity at\u00a0[latex] g=2. [\/latex] We plot the graphs for the different formulas on a common set of axes, making sure each formula is applied on its proper domain.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_506\" align=\"aligncenter\" width=\"351\"]<img class=\" wp-image-506\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-24-300x207.jpg\" alt=\"\" width=\"351\" height=\"242\" \/> Figure 24[\/caption]\r\n\r\n<\/details><\/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 piecewise function, sketch a graph.<\/strong>\r\n<ol>\r\n \t<li>Indicate on the <em data-effect=\"italics\">x<\/em>-axis the boundaries defined by the intervals on each piece of the domain.<\/li>\r\n \t<li>For each piece of the domain, graph on that interval using the corresponding equation pertaining to that piece. Do not graph two functions over one interval because it would violate the criteria of 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 13: Graphing a Piecewise Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSketch a graph of the function.\r\n<p style=\"text-align: center;\">[latex] f(x) = \\begin{cases}x^2 &amp; \\text{if} &amp; x\\le1 \\\\3 &amp; \\text{if} &amp; 1&lt; x\\le2 \\\\ x &amp; \\text{if} &amp; x&gt; 2\\end{cases} [\/latex]<\/p>\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Each of the component functions is from our library of toolkit functions, so we know their shapes. We can imagine graphing each function and then limiting the graph to the indicated domain. At the endpoints of the domain, we draw open circles to indicate where the endpoint is not included because of a less-than or greater-than inequality; we draw a closed circle where the endpoint is included because of a less-than-or-equal-to or greater-than-or-equal-to inequality.\r\n\r\nFigure 25 shows the three components of the piecewise function graphed on separate coordinate systems.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_507\" align=\"aligncenter\" width=\"375\"]<img class=\"wp-image-507\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-25-300x88.jpg\" alt=\"\" width=\"375\" height=\"110\" \/> Figure 25 (a) [latex] f(x)=x^2 \\ \\text{if} \\ x\\le1 [\/latex] (b) [latex] f(x)=3 \\ \\text{if} \\ 1&lt; x\\le2 [\/latex] (c) [latex] f(x)=x \\ \\text{if} \\ x&gt;2 [\/latex][\/caption]Now that we have sketched each piece individually, we combine them in the same coordinate plane. See Figure 26.\r\n\r\n[caption id=\"attachment_508\" align=\"aligncenter\" width=\"358\"]<img class=\" wp-image-508\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-26-300x247.jpg\" alt=\"\" width=\"358\" height=\"295\" \/> Figure 26[\/caption]\r\n<h3>Analysis<\/h3>\r\nNote that the graph does pass the vertical line test even at\u00a0[latex] x=1 [\/latex] and\u00a0[latex] x=2 [\/latex] because the points\u00a0[latex] (1,3) [\/latex] and\u00a0[latex] (2, 2) [\/latex] are not part of the graph of the function, though\u00a0[latex] (1, 1) [\/latex] and\u00a0[latex] (2, 3) [\/latex] are.\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\nGraph the following piecewise function.\r\n<p style=\"text-align: center;\">[latex] f(x) = \\begin{cases}x^3 &amp; \\text{if} &amp; x&lt; -1 \\\\-2 &amp; \\text{if} &amp; -1&lt; x&lt; 4 \\\\ \\sqrt{x} &amp; \\text{if} &amp; x&gt; 4\\end{cases} [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_01_02_06\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137692562\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<div id=\"fs-id1165137433350\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\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: Can more than one formula from a piecewise function be applied to a value in the domain?<\/strong>\r\n\r\n<em>A: No. Each value corresponds to one equation in a piecewise formula.<\/em>\r\n\r\n<\/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 these online resources for additional instruction and practice with domain and range.\r\n<ul>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=lj_JB8sfyIM\">Domain and Range of Square Root Functions<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=FtJRstFMdhA\">Determining Domain and Range<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=8jrkzZy04BQ\">Find Domain and Range Given the Graph<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=GPBq18fCEv4\">Find Domain and Range Given a Table<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=xOsYVyjTM0Q\">Find Domain and Range Given Points on a Coordinate Plane<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/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\">3.2 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1165135176628\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1165135172218\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1165137665109\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135245908\" 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-id1165137665109-solution\">1<\/a><span class=\"os-divider\">. <\/span>Why does the domain differ for different functions?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135440209\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135533141\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>How do we determine the domain of a function defined by an equation?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137635386\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135390940\" 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-id1165137635386-solution\">3<\/a><span class=\"os-divider\">. <\/span>Explain why the domain of\u00a0[latex] f(x)=\\sqrt[3]{x} [\/latex] is different from the domain of [latex] f(x)=\\sqrt{x} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134042454\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134042457\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134211324\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137446310\" 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-id1165134211324-solution\">5<\/a><span class=\"os-divider\">. <\/span>How do you graph a piecewise function?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165137771069\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id1165137408926\">For the following exercises, find the domain of each function using interval notation.<\/p>\r\n\r\n<div id=\"fs-id1165137833819\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137833821\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=-2x(x-1)(x-2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137854912\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134312130\" 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-id1165137854912-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=5-2x^2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135512534\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135512537\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=3\\sqrt{x-2} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137473385\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137473388\" 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-id1165137473385-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=3-\\sqrt{6-2x} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135192268\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135192270\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\sqrt{4-3x} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137629066\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137483196\" 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-id1165137629066-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\sqrt{x^2+4} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137807107\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137551129\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\sqrt[3]{1-2x} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134259277\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134259279\" 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-id1165134259277-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\sqrt[3]{x-1} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135503751\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134156030\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex] f(x0=\\frac{9}{x-6} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133276237\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133276240\" 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-id1165133276237-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{3x+1}{4x+2} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137810520\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137810522\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{\\sqrt{x+4}}{x-4} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135548992\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135634123\" 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-id1165135548992-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{x-3}{x^2+9x-22} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135593402\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135593404\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{1}{x^2-x-6} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135191342\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134284474\" 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-id1165135191342-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{2x^3-250}{x^2-2x-15} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137921795\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137921797\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{5}{\\sqrt{x-3}} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137476914\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137476916\" 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-id1165137476914-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{2x+1}{\\sqrt{5-x}} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135185292\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137640755\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{\\sqrt{x-4}}{\\sqrt{x-6}} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135252252\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137611840\" 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-id1165135252252-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{\\sqrt{x-6}}{\\sqrt{x-4}} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137601712\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137601714\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{x}{x} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137628472\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137651574\" 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-id1165137628472-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{x^2-9x}{x^2-81} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137469452\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137469454\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span>Find the domain of the function\u00a0[latex] f(x)=\\sqrt{2x^3-50x} [\/latex] by:\r\n\r\n(a) using algebra.\r\n\r\n(b) graphing the function in the radicand and determining intervals on the <em data-effect=\"italics\">x<\/em>-axis for which the radicand is nonnegative.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165137580833\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphical<\/h3>\r\n<p id=\"fs-id1165135186809\">For the following exercises, write the domain and range of each function using interval notation.<\/p>\r\n\r\n<div id=\"fs-id1165135168172\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137647479\" 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-id1165135168172-solution\">27<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165137891294\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from (2, 8].\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1185\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.27-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135160181\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135160183\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165137837830\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from [4, 8).\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1186\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.28-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137723404\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137809982\" 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-id1165137723404-solution\">29<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165137733767\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from [-4, 4].\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1187\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.29-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137590678\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134168421\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165137837060\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from [2, 6].\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1188\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.30-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137737326\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137737328\" 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-id1165137737326-solution\">31<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165134129572\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from [-5, 3).\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1189\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.31-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137404973\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137404975\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165134305418\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from [-3, 2).\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1190\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.32-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137544188\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137437269\" 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-id1165137544188-solution\">33<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165137447903\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from (-infinity, 2].\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1191\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.33-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135176309\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134323791\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165135192955\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from [-4, infinity).\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1192\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.34-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137642580\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137642582\" 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-id1165137642580-solution\">35<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165134482733\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from [-6, -1\/6]U[1\/6, 6]\/.\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1193\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.35-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137442385\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137812572\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165137645308\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from (-2.5, infinity).\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1194\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.36-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137851981\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137851983\" 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-id1165137851981-solution\">37<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165137602824\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from [-3, infinity).\" data-display=\"block\"><img class=\"alignnone size-medium wp-image-1195\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.37-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" \/><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137785119\">For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.<\/p>\r\n\r\n<div id=\"fs-id1165137462167\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137408525\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex] f(x) = \\begin{cases}x+1 &amp; \\text{if} &amp; x&lt;-2 \\\\ -2x-3 &amp; \\text{if} &amp; x\\ge-2 \\end{cases} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137562309\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134328320\" 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-id1165137562309-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex] f(x) = \\begin{cases}2x-1 &amp;\u00a0 \\text{if} &amp; x&lt;1 \\\\ 1+x &amp; \\text{if} &amp; x\\ge1 \\end{cases} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137628033\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137658060\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex] f(x) = \\begin{cases}x+1 &amp; \\text{if} &amp; x&lt;0 \\\\ x-1 &amp; \\text{if} &amp; x&gt;0 \\end{cases} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135641679\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135641681\" 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-id1165135641679-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex] f(x) = \\begin{cases}3 &amp; \\text{if} &amp; x&lt;0 \\\\ \\sqrt{x} &amp; \\text{if} &amp; x\\ge0 \\end{cases} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135192719\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135192721\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex] f(x) = \\begin{cases}x^2 &amp; \\text{if} &amp; x&lt;0 \\\\ 1-x &amp; \\text{if} &amp; x&gt;0 \\end{cases} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137594981\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135210029\" 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-id1165137594981-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex] f(x) = \\begin{cases}x^2 &amp; \\text{if} &amp; x&lt;0 \\\\ x+2 &amp; \\text{if} &amp; x\\ge0 \\end{cases} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137571389\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137433000\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex] f(x) = \\begin{cases}x+1 &amp; \\text{if} &amp; x&lt;1 \\\\ x^3 &amp; \\text{if} &amp; x\\ge0 \\end{cases} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137407891\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137554125\" 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-id1165137407891-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex] f(x) = \\begin{cases}|x| &amp; \\text{if} &amp; x&lt;2 \\\\ 1 &amp; \\text{if} &amp; x\\ge2 \\end{cases} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165134118450\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Numeric<\/h3>\r\n<p id=\"fs-id1165135188383\">For the following exercises, given each function\u00a0[latex] f [\/latex] evaluate\u00a0[latex] f(-3), f(-2), f(-1), [\/latex] and [latex] f(0). [\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165137471865\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137471867\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex] f(x) = \\begin{cases}x+1 &amp; \\text{if} &amp; x&lt;-2 \\\\ -2x-3 &amp; \\text{if} &amp; x\\ge-2 \\end{cases} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134122954\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134122956\" 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-id1165134122954-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex] f(x) = \\begin{cases}1 &amp; \\text{if} &amp; x\\le-3 \\\\ 0 &amp; \\text{if} &amp; x&gt;-3 \\end{cases} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137556768\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137423742\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex] f(x) = \\begin{cases}-2x^2+3 &amp; \\text{if} &amp; x\\le-1 \\\\ 5x-7 &amp; \\text{if} &amp; x&gt;-1 \\end{cases} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137469026\">For the following exercises, given each function\u00a0[latex] f [\/latex] evaluate\u00a0[latex] f(-1), f(0), f(2), [\/latex] and [latex] f(4). [\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165134380351\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134380353\" 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-id1165134380351-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex] f(x) = \\begin{cases}7x+3 &amp; \\text{if} &amp; x&lt;0 \\\\ 7x+6 &amp; \\text{if} &amp; x\\ge0 \\end{cases} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137693713\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137679373\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex] f(x) = \\begin{cases}x^2-2 &amp; \\text{if} &amp; x&lt;2 \\\\ 4+|x-5| &amp; \\text{if} &amp; x\\ge2 \\end{cases} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137715004\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137715006\" 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-id1165137715004-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex] f(x) = \\begin{cases}5x &amp; \\text{if} &amp; x&lt;0 \\\\ 3 &amp; \\text{if} &amp; x\\le0\\le3 \\\\ x^2 &amp; \\text{if} &amp; x&gt;3 \\end{cases} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137837869\">For the following exercises, write the domain for the piecewise function in interval notation.<\/p>\r\n\r\n<div id=\"fs-id1165137837872\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135341427\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex] f(x) = \\begin{cases}x+1 &amp; \\text{if} &amp; x&lt;-2 \\\\ -2x-3 &amp; \\text{if} &amp; x\\ge-2\\end{cases} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137704661\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137704664\" 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-id1165137704661-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex] f(x) = \\begin{cases}x^2-2 &amp; \\text{if} &amp; x&lt;1 \\\\ -x^2+2 &amp; \\text{if} &amp; x&gt;1 \\end{cases} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137772429\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137772431\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span> [latex] f(x) = \\begin{cases}2x-3 &amp; \\text{if} &amp; x&lt;0 \\\\ -3x^2 &amp; \\text{if} &amp; x\\ge2 \\end{cases} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165135194497\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<div id=\"fs-id1165137780865\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137780867\" 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-id1165137780865-solution\">55<\/a><span class=\"os-divider\">. <\/span>Graph\u00a0[latex] y=\\frac{1}{x^2} [\/latex] on the viewing window\u00a0[latex] [-0.5, -0.1] [\/latex] and\u00a0[latex] [0.1, 0.5]. [\/latex] Determine the corresponding range for the viewing window. Show the graphs.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165131911953\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137842479\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>Graph\u00a0[latex] y=\\frac{1}{x} [\/latex] on the viewing window\u00a0[latex] [-0.5, -0.1] [\/latex] and\u00a0[latex] [0.1, 0.5]. [\/latex] Determine the corresponding range for the viewing window. Show the graphs.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165137733672\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extension<\/h3>\r\n<div id=\"fs-id1165137442197\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133221851\" 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-id1165137442197-solution\">57<\/a><span class=\"os-divider\">. <\/span>Suppose the range of a function\u00a0[latex] f [\/latex] is\u00a0[latex] [-5, 8]. [\/latex] What is the range of [latex] |f(x)|? [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137679047\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137679049\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>Create a function in which the range is all nonnegative real numbers.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135209378\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135209380\" 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-id1165135209378-solution\">59<\/a><span class=\"os-divider\">. <\/span>Create a function in which the domain is [latex] x&gt;2. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165137832031\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<div id=\"fs-id1165135511303\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135511305\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span>A physics student at CCA is doing an experiment which tracks the motions of a rock in the air. The height\u00a0[latex] h [\/latex] of the rock is a function of the time\u00a0[latex] t [\/latex] it is in the air. The height in feet for\u00a0[latex] t [\/latex] seconds is given by the function\u00a0[latex] h(t)=-16t^2+96t. [\/latex] What is the domain of the function? What does the domain mean in the context of the problem?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137406705\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137406708\" 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-id1165137406705-solution\">61<\/a><span class=\"os-divider\">. <\/span>The cost in dollars of making\u00a0[latex] x [\/latex] items is given by the function [latex] C(x)=10x+500. [\/latex]\r\n\r\n(a) The fixed cost is determined when zero items are produced. Find the fixed cost for this item.\r\n\r\n(b) What is the cost of making 25 items?\r\n\r\n(c) Suppose the maximum cost allowed is $1500. What are the domain and range of the cost function, [latex] C(x)? [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>\r\n<div data-type=\"footnote-refs\"><\/div>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_562c3737-a93d-458c-98c0-a04f442f13bd\" 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>Find the domain of a function defined by an equation.<\/li>\n<li>Graph piecewise-defined functions.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<section>\n<ul id=\"list-00001\"><\/ul>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137404978\">Horror and thriller movies are both popular and, very often, extremely profitable. When big-budget actors, shooting locations, and special effects are included, however, studios count on even more viewership to be successful. Consider five major thriller\/horror entries from the early 2000s\u2014<em data-effect=\"italics\">I am Legend<\/em>, <em data-effect=\"italics\">Hannibal<\/em>, <em data-effect=\"italics\">The Ring<\/em>, <em data-effect=\"italics\">The Grudge<\/em>, and <em data-effect=\"italics\">The Conjuring<\/em>. Figure 1 shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the <span id=\"term-00007\" class=\"no-emphasis\" data-type=\"term\">domain<\/span> and range. In this section, we will investigate methods for determining the domain and range of functions such as these.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_480\" aria-describedby=\"caption-attachment-480\" style=\"width: 539px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-480\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-1-300x114.jpg\" alt=\"\" width=\"539\" height=\"205\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-1-300x114.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-1-1024x390.jpg 1024w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-1-768x292.jpg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-1-65x25.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-1-225x86.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-1-350x133.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-1.jpg 1096w\" sizes=\"auto, (max-width: 539px) 100vw, 539px\" \/><figcaption id=\"caption-attachment-480\" class=\"wp-caption-text\">Figure 1.<\/figcaption><\/figure>\n<section id=\"fs-id1165135193832\" data-depth=\"1\">\n<p data-type=\"title\"><span class=\"os-caption\">(Based on data compiled by www.the-numbers.com.<a class=\"footnote\" title=\"The Numbers: Where Data and the Movie Business Meet. \u201cBox Office History for Horror Movies.\u201d http:\/\/www.the-numbers.com\/market\/genre\/Horror. Accessed 3\/24\/2014\" id=\"return-footnote-133-1\" href=\"#footnote-133-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>)<\/span><\/p>\n<h2 data-type=\"title\">Finding the Domain of a Function Defined by an Equation<\/h2>\n<p id=\"fs-id1165135445896\">In Functions and Function Notation, we were introduced to the concepts of <span id=\"term-00008\" class=\"no-emphasis\" data-type=\"term\">domain and range<\/span>. In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.<\/p>\n<p id=\"fs-id1165135453892\">We can visualize the domain as a \u201cholding area\u201d that contains \u201craw materials\u201d for a \u201cfunction machine\u201d and the range as another \u201cholding area\u201d for the machine\u2019s products. See Figure 2.<\/p>\n<p>&nbsp;<\/p>\n<\/section>\n<figure id=\"attachment_481\" aria-describedby=\"caption-attachment-481\" style=\"width: 670px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-481\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-2-300x141.jpg\" alt=\"\" width=\"670\" height=\"315\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-2-300x141.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-2-65x31.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-2-225x106.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-2-350x164.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-2.jpg 415w\" sizes=\"auto, (max-width: 670px) 100vw, 670px\" \/><figcaption id=\"caption-attachment-481\" class=\"wp-caption-text\">Figure 2<\/figcaption><\/figure>\n<section data-depth=\"1\">\n<p id=\"fs-id1165137761714\">We can write the <span id=\"term-00009\" class=\"no-emphasis\" data-type=\"term\">domain and range<\/span> in <span id=\"term-00010\" data-type=\"term\">interval notation<\/span>, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, they would need to express the interval that is more than 0 and less than or equal to 100 and write [latex](0, 100].[\/latex] We will discuss interval notation in greater detail later.<\/p>\n<p id=\"fs-id1165135320406\">Let\u2019s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an odd root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function\u2019s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.<\/p>\n<p id=\"fs-id1165137552233\">Before we begin, let us review the conventions of interval notation:<\/p>\n<ul id=\"fs-id1165135673417\">\n<li>The smallest number from the interval is written first.<\/li>\n<li>The largest number in the interval is written second, following a comma.<\/li>\n<li>Parentheses, ( or ), are used to signify that an endpoint value is not included, called exclusive.<\/li>\n<li>Brackets, [ or ], are used to indicate that an endpoint value is included, called inclusive.<\/li>\n<\/ul>\n<p id=\"fs-id1165137807384\">See Figure 3 for a summary of interval notation.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_482\" aria-describedby=\"caption-attachment-482\" style=\"width: 588px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-482\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-3-300x278.jpg\" alt=\"\" width=\"588\" height=\"545\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-3-300x278.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-3-768x713.jpg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-3-65x60.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-3-225x209.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-3-350x325.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-3.jpg 975w\" sizes=\"auto, (max-width: 588px) 100vw, 588px\" \/><figcaption id=\"caption-attachment-482\" class=\"wp-caption-text\">Figure 3<\/figcaption><\/figure>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 1: Finding the Domain of a Function as a Set of Ordered Pairs<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the <span id=\"term-00011\" class=\"no-emphasis\" data-type=\"term\">domain<\/span> of the following function: [latex]\\{(2, 10), (3, 10), (4, 20), (5, 30), (6, 40)\\}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>First identify the input values. The input value is the first coordinate in an ordered pair. There are no restrictions, as the ordered pairs are simple listed. The domain is the best of the first coordinates of the ordered pairs.<\/p>\n<p style=\"text-align: center;\">[latex]\\{2, 3, 4, 5, 6\\}[\/latex]<\/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>Find the domain of the function: [latex]\\{(-5, 4), (0, 0), (5, -4), (10, -8), (15, -12)\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\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 written in equation form, find the domain.<\/strong><\/p>\n<ol>\n<li>Identify the input values.<\/li>\n<li>Identify any restrictions on the input and exclude those values from the domain.<\/li>\n<li>Write the domain in interval form, if possible.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2: Finding the Domain of a Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the domain of the function [latex]f(x)=x^2-1.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The input value, shown by the variable\u00a0[latex]x[\/latex] in the equation, is squared and then the result is lowered by one. Any real number may be squared and then lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers.<\/p>\n<p>In interval dorm, the domain of\u00a0[latex]f[\/latex] is [latex](-\\infty, \\infty).[\/latex]<\/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>Find the domain of the function: [latex]f(x)=5-x+x^2.[\/latex]<\/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><strong>Given a function written in an equation form that includes a fraction, find the domain.<\/strong><\/p>\n<ol>\n<li>Identify the input values.<\/li>\n<li>Identify any restrictions on the input. If there is a denominator in the function&#8217;s formula, set the denominator equal to zero and solve for x. If the function&#8217;s formula contains an even root, set the radicand greater than or equal to 0, and then solve.<\/li>\n<li>Write the domain in interval dorm, making sure to exclude any restricted values from the domain.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Finding the Domain of a Function Involving a Denominator<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the domain of the function [latex]f(x)=\\frac{x+1}{2-x}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for [latex]x.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{ll} 2-x & =0 \\\\ -x & =-2 \\\\ x & =2 \\end{array}[\/latex]<\/p>\n<p>Now, we will exclude 2 from the domain. The answers are all real numbers where [latex]x<2[\/latex] or [latex]x>2[\/latex] as shown in Figure 4. We can use a symbol known as the union, [latex]\\cup[\/latex] to combine the two sets. In interval notation, we write the solution: [latex](-\\infty, 2)\\cup(2, \\infty)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_483\" aria-describedby=\"caption-attachment-483\" style=\"width: 431px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-483\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-4-300x101.jpg\" alt=\"\" width=\"431\" height=\"145\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-4-300x101.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-4-65x22.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-4-225x76.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-4-350x118.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-4.jpg 487w\" sizes=\"auto, (max-width: 431px) 100vw, 431px\" \/><figcaption id=\"caption-attachment-483\" class=\"wp-caption-text\">Figure 4<\/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 #3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the domain of the function: [latex]f(x)=\\frac{1+4x}{2x-1}[\/latex]<\/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><strong>Given a function written in equation form including an even root, find the domain.<\/strong><\/p>\n<ol>\n<li>Identify the input values.<\/li>\n<li>Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for x.<\/li>\n<li>The solution(s) are the domain of the function. If possible, write the answer in interval form.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4: Find the Domain of a Function with an Even Root<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the domain of the function [latex]f(x)=\\sqrt{7-x}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.<\/p>\n<p>Set the radicand greater than or equal to zero and solve for [latex]x.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\u00a0\\begin{array}{ll} 7-x & \\ge0 \\\\ -x & \\ge-7 \\\\ x & \\le7 \\end{array}[\/latex]<\/p>\n<p>Now, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal to 7, or [latex](-\\infty, 7].[\/latex]<\/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>Find the domain of the function [latex]f(x)=\\sqrt{5+2x}.[\/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: Can there be functions in which the domain and range do not intersect at all?<\/strong><\/p>\n<p><em>A: Yes. For example, the function\u00a0[latex]f(x)=-\\frac{1}{\\sqrt{x}}[\/latex] <\/em><em>has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a functions inputs and output can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.<\/em><\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165137677916\" data-depth=\"1\">\n<h2 data-type=\"title\">Using Notations to Specify Domain and Range<\/h2>\n<p id=\"fs-id1165137410091\">In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in <strong><span id=\"term-00015\" data-type=\"term\">set-builder notation<\/span><\/strong>. For example,\u00a0[latex]\\{x|10\\le x\\le30\\}[\/latex] describes the behavior of\u00a0[latex]x[\/latex] in set-builder notation. The braces\u00a0[latex]\\{\\}[\/latex] are read as \u201cthe set of,\u201d and the vertical bar | is read as \u201csuch that,\u201d so we would read\u00a0[latex]\\{x|10\\le x\\le30\\}[\/latex] as \u201cthe set of <em data-effect=\"italics\">x<\/em>-values such that 10 is less than or equal to\u00a0[latex]x[\/latex] and\u00a0[latex]x[\/latex] is less than 30.\u201d<\/p>\n<p id=\"fs-id1165135207589\">Figure 5 compares inequality notation, set-builder notation, and interval notation.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_484\" aria-describedby=\"caption-attachment-484\" style=\"width: 600px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-484\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-5-300x222.jpg\" alt=\"\" width=\"600\" height=\"444\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-5-300x222.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-5-1024x759.jpg 1024w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-5-768x569.jpg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-5-1536x1138.jpg 1536w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-5-65x48.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-5-225x167.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-5-350x259.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-5.jpg 1871w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><figcaption id=\"caption-attachment-484\" class=\"wp-caption-text\">Figure 5<\/figcaption><\/figure>\n<p id=\"fs-id1165137911528\">To combine two intervals using inequality notation or set-builder notation, we use the word \u201cor.\u201d As we saw in earlier examples, we use the union symbol,\u00a0[latex]\\cup[\/latex]\u00a0to combine two unconnected intervals. For example, the union of the sets\u00a0[latex]\\{2, 3, 5\\}[\/latex] and\u00a0[latex]\\{4, 6\\}[\/latex] is the set\u00a0[latex]\\{2, 3, 4, 5, 6\\}[\/latex] It is the set of all elements that belong to one <em data-effect=\"italics\">or<\/em> the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is<\/p>\n<p><math display=\"block\"><\/math><\/p>\n<p style=\"text-align: center;\">[latex]\\{x| \\ \\ \\ |x| \\ge3\\}=(-\\infty, -3]\\cup[3, \\infty)[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Set-Builder Notation and Interval Notation<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Set-builder notation<\/strong> is a method of specifying a set of elements that satisfy a certain condition. It takes the form\u00a0[latex]\\{x|\\hspace{0.5em}\\text{statement about}\\hspace{0.25em}x\\}[\/latex] which is read as, &#8220;the set of all x such that the statement about x is true. For example,<\/p>\n<p style=\"text-align: center;\">[latex]\\{x|4< x\\le12\\}[\/latex]<\/p>\n<p style=\"text-align: left;\"><strong>Interval notation<\/strong> is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. For example,<\/p>\n<p style=\"text-align: center;\">[latex](4, 12][\/latex]<\/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><strong>Given a line graph, describe the set of values using interval notation.<\/strong><\/p>\n<ol>\n<li>Identify the intervals to be included in the set by determining where the heavy line overlays the real line.<\/li>\n<li>At the left end of each interval, use [ with each end value to be included in the set (solid dot) or ( for each excluded end value (open dot).<\/li>\n<li>At the right end of each interval, use ] with each end value to be included in the set (filled dot) or ) for each excluded end value (open dot).<\/li>\n<li>Use the union symbol\u00a0[latex]\\cup[\/latex] to combine all intervals into one set.<\/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: Describing Sets on the Real-Number Line<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Describe the intervals of values shown in Figure 6 using inequality notation, set-builder notation, and interval notation.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_485\" aria-describedby=\"caption-attachment-485\" style=\"width: 385px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-485\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-6-300x39.jpg\" alt=\"\" width=\"385\" height=\"50\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-6-300x39.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-6-65x9.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-6-225x30.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-6-350x46.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-6.jpg 418w\" sizes=\"auto, (max-width: 385px) 100vw, 385px\" \/><figcaption id=\"caption-attachment-485\" class=\"wp-caption-text\">Figure 6<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>To describe the values,\u00a0[latex]x,[\/latex] included in the intervals shown, we would say, &#8220;[latex]x[\/latex] is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.&#8221;<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 45px;\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"width: 50%; height: 15px; text-align: center;\"><strong>Inequality<\/strong><\/td>\n<td style=\"width: 50%; height: 15px; text-align: center;\">[latex]1\\le x \\le3\u00a0 \\ \\text{or} \\ x>5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 50%; height: 15px; text-align: center;\"><strong>Set-builder notation<\/strong><\/td>\n<td style=\"width: 50%; height: 15px; text-align: center;\">[latex]\\{x|1 \\le x\\le3 \\ \\text{or} \\ x>5\\}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 50%; height: 15px; text-align: center;\"><strong>Interval notation<\/strong><\/td>\n<td style=\"width: 50%; height: 15px; text-align: center;\">[latex][1, 3]\\cup(5, \\infty)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Remember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. Using a parenthesis means the boundary is not included in the set.<\/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 #5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given Figure 7, specify the graphed set in<\/p>\n<p>(a) words<\/p>\n<p>(b) set-builder notation<\/p>\n<p>(c) interval notation<\/p>\n<figure id=\"attachment_486\" aria-describedby=\"caption-attachment-486\" style=\"width: 393px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-486\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-7-300x45.jpg\" alt=\"\" width=\"393\" height=\"59\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-7-300x45.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-7-65x10.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-7-225x34.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-7-350x52.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-7.jpg 369w\" sizes=\"auto, (max-width: 393px) 100vw, 393px\" \/><figcaption id=\"caption-attachment-486\" class=\"wp-caption-text\">Figure 7<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137653855\" data-depth=\"1\">\n<h2 data-type=\"title\">Finding Domain and Range from Graphs<\/h2>\n<p id=\"fs-id1165135161404\">Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the <em data-effect=\"italics\">x<\/em>-axis. The range is the set of possible output values, which are shown on the <em data-effect=\"italics\">y<\/em>-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See Figure 8.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_487\" aria-describedby=\"caption-attachment-487\" style=\"width: 555px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-487\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-8-230x300.jpg\" alt=\"\" width=\"555\" height=\"724\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-8-230x300.jpg 230w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-8-65x85.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-8-225x293.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-8-350x456.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-8.jpg 362w\" sizes=\"auto, (max-width: 555px) 100vw, 555px\" \/><figcaption id=\"caption-attachment-487\" class=\"wp-caption-text\">Figure 8<\/figcaption><\/figure>\n<p id=\"fs-id1165137597994\">We can observe that the graph extends horizontally from\u00a0[latex]-5[\/latex] to the right without bound, so the domain is\u00a0[latex][-5, \\infty).[\/latex] The vertical extent of the graph is all range values\u00a0[latex]5[\/latex] and below, so the range is\u00a0[latex](-\\infty, 5].[\/latex] Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 6: Finding Domain and Range from a Graph<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the domain and range of the function f whose graph is shown in Figure 9.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_488\" aria-describedby=\"caption-attachment-488\" style=\"width: 358px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-488\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-9-300x208.jpg\" alt=\"\" width=\"358\" height=\"248\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-9-300x208.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-9-65x45.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-9-225x156.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-9-350x243.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-9.jpg 488w\" sizes=\"auto, (max-width: 358px) 100vw, 358px\" \/><figcaption id=\"caption-attachment-488\" 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>We can observe that the horizontal extent of the graph is -3 to 1, so the domain of\u00a0[latex]f[\/latex]\u00a0is [latex](-3, 1].[\/latex]<\/p>\n<p>The vertical extent of the graph is 0 to -4, so the range is\u00a0[latex][-4, 0].[\/latex] See Figure 10.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_489\" aria-describedby=\"caption-attachment-489\" style=\"width: 508px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-489\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-10-300x224.jpg\" alt=\"\" width=\"508\" height=\"379\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-10-300x224.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-10-65x48.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-10-225x168.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-10-350x261.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-10.jpg 488w\" sizes=\"auto, (max-width: 508px) 100vw, 508px\" \/><figcaption id=\"caption-attachment-489\" 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\">Example 7: Finding Domain and Range from a Graph of iPhone Sales<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the domain and range of the function [latex]f[\/latex] whose graph is shown in Figure 11.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_492\" aria-describedby=\"caption-attachment-492\" style=\"width: 550px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-492\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-300x211.png\" alt=\"\" width=\"550\" height=\"387\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-300x211.png 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-768x539.png 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-65x46.png 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-225x158.png 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3-350x246.png 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.3.png 1021w\" sizes=\"auto, (max-width: 550px) 100vw, 550px\" \/><figcaption id=\"caption-attachment-492\" class=\"wp-caption-text\">Figure 11<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The input quantity along the horizontal axis is &#8220;years&#8221; which we represent with the variable [latex]t[\/latex] for time. The output quantity is &#8220;billions of dollars per year,&#8221; which we represent with the variable [latex]d[\/latex] for dollars. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as [latex]2019\\le t\\le2023[\/latex] and the range as approximately\u00a0[latex]137.7\\le d\\le205.5[\/latex]<\/p>\n<p>In interval notation, the domain is\u00a0[latex][2019, 2023][\/latex] and the range is\u00a0[latex][137.7, 205.5].[\/latex] For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.<\/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 #6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given Figure 12, identify the domain and range using interval notation.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_493\" aria-describedby=\"caption-attachment-493\" style=\"width: 468px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-493\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-12-300x204.jpg\" alt=\"\" width=\"468\" height=\"318\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-12-300x204.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-12-65x44.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-12-225x153.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-12-350x238.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-12.jpg 372w\" sizes=\"auto, (max-width: 468px) 100vw, 468px\" \/><figcaption id=\"caption-attachment-493\" class=\"wp-caption-text\">Figure 12<\/figcaption><\/figure>\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: Can a function&#8217;s domain and range be the same?<\/strong><\/p>\n<p><em>A: Yes. For example, the domain and range of the cube root function are both the set of all real numbers.<\/em><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165134384565\" data-depth=\"1\">\n<h2 data-type=\"title\">Finding Domains and Ranges of the Toolkit Functions<\/h2>\n<p id=\"fs-id1165137419914\">We will now return to our set of toolkit functions to determine the domain and range of each.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_494\" aria-describedby=\"caption-attachment-494\" style=\"width: 370px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-494\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-13-259x300.jpg\" alt=\"\" width=\"370\" height=\"429\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-13-259x300.jpg 259w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-13-65x75.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-13-225x260.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-13.jpg 307w\" sizes=\"auto, (max-width: 370px) 100vw, 370px\" \/><figcaption id=\"caption-attachment-494\" class=\"wp-caption-text\">Figure 13. For the <strong>constant function<\/strong> [latex] f(x)=c, [\/latex] the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant [latex] c, [\/latex] so the range is the set [latex] \\{c\\} [\/latex] that contains this single element. In interval notation, this is written as [latex] [c, c], [\/latex] the interval that both begins and ends with [latex] c. [\/latex]<\/figcaption><\/figure>\n<figure id=\"attachment_495\" aria-describedby=\"caption-attachment-495\" style=\"width: 273px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-495\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-14-264x300.jpg\" alt=\"\" width=\"273\" height=\"310\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-14-264x300.jpg 264w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-14-65x74.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-14-225x256.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-14.jpg 307w\" sizes=\"auto, (max-width: 273px) 100vw, 273px\" \/><figcaption id=\"caption-attachment-495\" class=\"wp-caption-text\">Figure 14. For the <strong>identity function<\/strong> [latex] f(x)=x, [\/latex] there is no restriction on [latex] x. [\/latex] Both the domain and range are the set of all real numbers.<\/figcaption><\/figure>\n<p>\u00a0<\/p>\n<figure id=\"attachment_496\" aria-describedby=\"caption-attachment-496\" style=\"width: 288px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-496\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-15-262x300.jpg\" alt=\"\" width=\"288\" height=\"330\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-15-262x300.jpg 262w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-15-65x74.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-15-225x257.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-15.jpg 307w\" sizes=\"auto, (max-width: 288px) 100vw, 288px\" \/><figcaption id=\"caption-attachment-496\" class=\"wp-caption-text\">Figure 15. For the <strong>absolute value function<\/strong> [latex] f(x)=|x|, [\/latex] there is no restriction on [latex] x. [\/latex] However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0.<\/figcaption><\/figure>\n<p>\u00a0<\/p>\n<figure id=\"attachment_497\" aria-describedby=\"caption-attachment-497\" style=\"width: 318px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-497\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-16-300x290.jpg\" alt=\"\" width=\"318\" height=\"307\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-16-300x290.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-16-65x63.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-16-225x218.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-16.jpg 307w\" sizes=\"auto, (max-width: 318px) 100vw, 318px\" \/><figcaption id=\"caption-attachment-497\" class=\"wp-caption-text\">Figure 16. For the <strong>quadratic function<\/strong> [latex] f(x)=x^2 [\/latex] the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.<\/figcaption><\/figure>\n<p>\u00a0<\/p>\n<figure id=\"attachment_498\" aria-describedby=\"caption-attachment-498\" style=\"width: 278px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-498\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-17-262x300.jpg\" alt=\"\" width=\"278\" height=\"318\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-17-262x300.jpg 262w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-17-65x74.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-17-225x257.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-17.jpg 307w\" sizes=\"auto, (max-width: 278px) 100vw, 278px\" \/><figcaption id=\"caption-attachment-498\" class=\"wp-caption-text\">Figure 17. For the <strong>cubic function<\/strong> [latex] f(x)=x^3 [\/latex] the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.<\/figcaption><\/figure>\n<p>\u00a0<\/p>\n<figure id=\"attachment_499\" aria-describedby=\"caption-attachment-499\" style=\"width: 287px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-499\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-18-259x300.jpg\" alt=\"\" width=\"287\" height=\"332\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-18-259x300.jpg 259w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-18-65x75.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-18-225x260.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-18.jpg 307w\" sizes=\"auto, (max-width: 287px) 100vw, 287px\" \/><figcaption id=\"caption-attachment-499\" class=\"wp-caption-text\">Figure 18. For the <strong>reciprocal function<\/strong> [latex] f(x)=\\frac{1}{x} [\/latex] we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also write [latex] \\{x|\\hspace{0.5em}x\\not=0\\} [\/latex] the set of all real numbers that are not zero.<\/figcaption><\/figure>\n<p>\u00a0<\/p>\n<figure id=\"attachment_500\" aria-describedby=\"caption-attachment-500\" style=\"width: 316px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-500\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-19-300x300.jpg\" alt=\"\" width=\"316\" height=\"316\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-19-300x300.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-19-150x150.jpg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-19-65x65.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-19-225x225.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-19.jpg 307w\" sizes=\"auto, (max-width: 316px) 100vw, 316px\" \/><figcaption id=\"caption-attachment-500\" class=\"wp-caption-text\">Figure 19. For the <strong>reciprocal squared function<\/strong> [latex] f(x)=\\frac{1}{x^2} [\/latex] we cannot divide by 0, so we must exclude 0 from the domain. There is also no [latex] x [\/latex] that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.<\/figcaption><\/figure>\n<p>\u00a0<\/p>\n<figure id=\"attachment_501\" aria-describedby=\"caption-attachment-501\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-501\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-20-300x297.jpg\" alt=\"\" width=\"310\" height=\"307\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-20-300x297.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-20-150x150.jpg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-20-65x64.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-20-225x223.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-20.jpg 307w\" sizes=\"auto, (max-width: 310px) 100vw, 310px\" \/><figcaption id=\"caption-attachment-501\" class=\"wp-caption-text\">Figure 20. For the <strong>square root function<\/strong> [latex] f(x)=\\sqrt{x}, [\/latex] we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number x is defined to be positive, even though the square of the negative number [latex] -\\sqrt{x} [\/latex] also gives us [latex] x. [\/latex]<\/figcaption><\/figure>\n<p><math display=\"inline\"><semantics><mrow><\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<figure id=\"attachment_502\" aria-describedby=\"caption-attachment-502\" style=\"width: 314px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-502\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-21-300x292.jpg\" alt=\"\" width=\"314\" height=\"306\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-21-300x292.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-21-65x63.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-21-225x219.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-21.jpg 307w\" sizes=\"auto, (max-width: 314px) 100vw, 314px\" \/><figcaption id=\"caption-attachment-502\" class=\"wp-caption-text\">Figure 21. For the <strong>cube root function<\/strong> [latex] f(x)=\\sqrt[3]{x}, [\/latex] the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function).<\/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 the formula for a function, determine the domain and range.<\/strong><\/p>\n<ol>\n<li>Exclude from the domain any input values that result in division by zero.<\/li>\n<li>Exclude from the domain any input values that have nonreal (or undefined) number outputs.<\/li>\n<li>Use the valid input values to determine the range of the output values.<\/li>\n<li>Look at the function graph and table values to confirm the actual function behavior.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 8: Finding the Domain and Range Using Toolkit Functions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the domain and range of [latex]f(x)=2x^3-x.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.<\/p>\n<p>The domain is\u00a0[latex](-\\infty, \\infty)[\/latex] and the range is also [latex](-\\infty, \\infty).[\/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 9: Finding the Domain and Range<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the domain and range of [latex]f(x)=\\frac{2}{x+1}.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We cannot evaluate the function at\u00a0[latex]-1[\/latex] because division by zero is undefined. The domain is\u00a0[latex](-\\infty, -1)  \\cup(01, \\infty).[\/latex] Because the function is never zero, we exclude 0 from the range. The range is [latex](-\\infty, 0)\\cup(0, \\infty).[\/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 10: Finding the Domain and Range<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the domain and range of [latex]f(x)=2\\sqrt{x+4}.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.<\/p>\n<p style=\"text-align: center;\">[latex]x+r\\ge0 \\ \\text{when} \\ x\\ge-4[\/latex]<\/p>\n<p>The domain of\u00a0[latex]f(x)[\/latex] is [latex][-4, \\infty).[\/latex]<\/p>\n<p>We then find the range. We know that\u00a0[latex]f(-4)=0,[\/latex] and the function value increases as [latex]x[\/latex] increases without any upper limit. We conclude that the range of\u00a0[latex]f[\/latex] is [latex][0, \\infty).[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>Figure 22 represents the function [latex]f.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_503\" aria-describedby=\"caption-attachment-503\" style=\"width: 318px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-503\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-22-300x218.jpg\" alt=\"\" width=\"318\" height=\"231\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-22-300x218.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-22-65x47.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-22-225x163.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-22-350x254.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-22.jpg 362w\" sizes=\"auto, (max-width: 318px) 100vw, 318px\" \/><figcaption id=\"caption-attachment-503\" class=\"wp-caption-text\">Figure 22<\/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 #7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the domain and range of [latex]f(x)=-\\sqrt{2-x}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135440477\" data-depth=\"1\">\n<h2 data-type=\"title\">Graphing Piecewise-Defined Functions<\/h2>\n<p id=\"fs-id1165137409262\">Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function\u00a0[latex]f(x)=|x|.[\/latex] With a domain of all real numbers and a range of values greater than or equal to 0, <span id=\"term-00018\" class=\"no-emphasis\" data-type=\"term\">absolute value<\/span> can be defined as the <span id=\"term-00019\" class=\"no-emphasis\" data-type=\"term\">magnitude<\/span>, or <span id=\"term-00020\" class=\"no-emphasis\" data-type=\"term\">modulus<\/span>, of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.<\/p>\n<p id=\"fs-id1165137558775\">If we input 0, or a positive value, the output is the same as the input.<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=x \\ \\text{if} \\ x\\ge0[\/latex]<\/p>\n<p id=\"fs-id1165137529947\">If we input a negative value, the output is the opposite of the input.<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=-x \\ \\text{if} \\ x<0[\/latex]<\/p>\n<p id=\"fs-id1165137863778\">Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A <strong><span id=\"term-00021\" data-type=\"term\">piecewise function<\/span><\/strong> is a function in which more than one formula is used to define the output over different pieces of the domain.<\/p>\n<p id=\"fs-id1165134042316\">We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain \u201cboundaries.\u201d For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income\u00a0[latex]S[\/latex] would be\u00a0[latex]0.1S[\/latex] if\u00a0[latex]S\\le\\$10,000[\/latex] and\u00a0[latex]\\$1000+0.2(S-\\$10,000)[\/latex] if [latex]S>\\$10,000.[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Piecewise Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A <strong>piecewise function<\/strong> is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:<\/p>\n<p style=\"text-align: center;\">[latex]f(x) = \\begin{cases}\\text{formula 1} & \\text{if } x \\text{ is in domain 1} \\\\\\text{formula 2} & \\text{if } x \\text{ is in domain 2} \\\\\\text{formula 3} & \\text{if } x \\text{ is in domain 3}\\end{cases}[\/latex]<\/p>\n<p>In piecewise notation, the absolute value function is<\/p>\n<p style=\"text-align: center;\">[latex]|x| = \\begin{cases}x \\quad \\text{ if } x \\geq 0 \\\\-x \\quad \\text{ if } x < 0\\end{cases}[\/latex]<\/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><strong>Given a piecewise function, write the formula and identify the domain for each interval.<\/strong><\/p>\n<ol>\n<li>Identify the intervals for which different rules apply.<\/li>\n<li>Determine formulas that describe how to calculate an output from an input in each interval.<\/li>\n<li>Use braces and if-statements to write the 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 11: Writing a Piecewise Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The Aurora History Museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a <span id=\"term-00022\" class=\"no-emphasis\" data-type=\"term\">function<\/span> relating the number of people,\u00a0[latex]n,[\/latex] to the cost, [latex]C.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Two different formulas will be needed. For [latex]n[\/latex]-values under 10,\u00a0[latex]C=5n.[\/latex] For values of [latex]n[\/latex] that are 10 or greater, [latex]C=50.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]C(n) = \\begin{cases}5n & \\text{if} & 0 < n < 10 \\\\50 & \\text{if} & n \\geq 10\\end{cases}[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>The function is represented in Figure 23. The graph is a diagonal line from\u00a0[latex]n=0[\/latex] to\u00a0[latex]n-10[\/latex] and a constant after that. In this example, the two formulas agree at the meeting point where\u00a0[latex]n=10,[\/latex] but not all piecewise functions have this property.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_504\" aria-describedby=\"caption-attachment-504\" style=\"width: 338px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-504\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-23-300x245.jpg\" alt=\"\" width=\"338\" height=\"276\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-23-300x245.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-23-65x53.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-23-225x184.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-23-350x286.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-23.jpg 360w\" sizes=\"auto, (max-width: 338px) 100vw, 338px\" \/><figcaption id=\"caption-attachment-504\" class=\"wp-caption-text\">Figure 23<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 12: Working with a Piecewise Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A cell phone company uses the function below to determine the cost,\u00a0[latex]C,[\/latex] in dollars for\u00a0[latex]g[\/latex] gigabytes of data transfer.<\/p>\n<p style=\"text-align: center;\">[latex]C(g) = \\begin{cases}25 & \\text{if} & 0 < g < 2 \\\\25 + 10(g - 2) & \\text{if} & g \\geq 2\\end{cases}[\/latex]<\/p>\n<p>Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>To find the cost of using 1.5 gigabytes of data,\u00a0[latex]C(1.5),[\/latex] we first look to see which part of the domain our input falls in. Because 1.5 is less than 2, we use the first formula.<\/p>\n<p style=\"text-align: center;\">[latex]C(1.5)=\\$25[\/latex]<\/p>\n<p>To find the cost of using 4 gigabytes of data,\u00a0[latex]C(4),[\/latex] we see that our input of 4 is greater than 2, so we use the second formula.<\/p>\n<p style=\"text-align: center;\">[latex]C(4)=25+10(4-2)=\\$45[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>The function is represented in Figure 24. We can see where the function changes from a constant to a shifted and stretched identity at\u00a0[latex]g=2.[\/latex] We plot the graphs for the different formulas on a common set of axes, making sure each formula is applied on its proper domain.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_506\" aria-describedby=\"caption-attachment-506\" style=\"width: 351px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-506\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-24-300x207.jpg\" alt=\"\" width=\"351\" height=\"242\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-24-300x207.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-24-65x45.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-24-225x155.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-24-350x241.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-24.jpg 490w\" sizes=\"auto, (max-width: 351px) 100vw, 351px\" \/><figcaption id=\"caption-attachment-506\" class=\"wp-caption-text\">Figure 24<\/figcaption><\/figure>\n<\/details>\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><strong>Given a piecewise function, sketch a graph.<\/strong><\/p>\n<ol>\n<li>Indicate on the <em data-effect=\"italics\">x<\/em>-axis the boundaries defined by the intervals on each piece of the domain.<\/li>\n<li>For each piece of the domain, graph on that interval using the corresponding equation pertaining to that piece. Do not graph two functions over one interval because it would violate the criteria of 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 13: Graphing a Piecewise Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Sketch a graph of the function.<\/p>\n<p style=\"text-align: center;\">[latex]f(x) = \\begin{cases}x^2 & \\text{if} & x\\le1 \\\\3 & \\text{if} & 1< x\\le2 \\\\ x & \\text{if} & x> 2\\end{cases}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Each of the component functions is from our library of toolkit functions, so we know their shapes. We can imagine graphing each function and then limiting the graph to the indicated domain. At the endpoints of the domain, we draw open circles to indicate where the endpoint is not included because of a less-than or greater-than inequality; we draw a closed circle where the endpoint is included because of a less-than-or-equal-to or greater-than-or-equal-to inequality.<\/p>\n<p>Figure 25 shows the three components of the piecewise function graphed on separate coordinate systems.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_507\" aria-describedby=\"caption-attachment-507\" style=\"width: 375px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-507\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-25-300x88.jpg\" alt=\"\" width=\"375\" height=\"110\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-25-300x88.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-25-1024x300.jpg 1024w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-25-768x225.jpg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-25-65x19.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-25-225x66.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-25-350x103.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-25.jpg 1169w\" sizes=\"auto, (max-width: 375px) 100vw, 375px\" \/><figcaption id=\"caption-attachment-507\" class=\"wp-caption-text\">Figure 25 (a) [latex] f(x)=x^2 \\ \\text{if} \\ x\\le1 [\/latex] (b) [latex] f(x)=3 \\ \\text{if} \\ 1&lt; x\\le2 [\/latex] (c) [latex] f(x)=x \\ \\text{if} \\ x&gt;2 [\/latex]<\/figcaption><\/figure>\n<p>Now that we have sketched each piece individually, we combine them in the same coordinate plane. See Figure 26.<\/p>\n<figure id=\"attachment_508\" aria-describedby=\"caption-attachment-508\" style=\"width: 358px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-508\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-26-300x247.jpg\" alt=\"\" width=\"358\" height=\"295\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-26-300x247.jpg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-26-65x53.jpg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-26-225x185.jpg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-26-350x288.jpg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2-fig-26.jpg 376w\" sizes=\"auto, (max-width: 358px) 100vw, 358px\" \/><figcaption id=\"caption-attachment-508\" class=\"wp-caption-text\">Figure 26<\/figcaption><\/figure>\n<h3>Analysis<\/h3>\n<p>Note that the graph does pass the vertical line test even at\u00a0[latex]x=1[\/latex] and\u00a0[latex]x=2[\/latex] because the points\u00a0[latex](1,3)[\/latex] and\u00a0[latex](2, 2)[\/latex] are not part of the graph of the function, though\u00a0[latex](1, 1)[\/latex] and\u00a0[latex](2, 3)[\/latex] are.<\/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 #8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Graph the following piecewise function.<\/p>\n<p style=\"text-align: center;\">[latex]f(x) = \\begin{cases}x^3 & \\text{if} & x< -1 \\\\-2 & \\text{if} & -1< x< 4 \\\\ \\sqrt{x} & \\text{if} & x> 4\\end{cases}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"os-note-body\">\n<div id=\"ti_01_02_06\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137692562\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<div id=\"fs-id1165137433350\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\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: Can more than one formula from a piecewise function be applied to a value in the domain?<\/strong><\/p>\n<p><em>A: No. Each value corresponds to one equation in a piecewise formula.<\/em><\/p>\n<\/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 these online resources for additional instruction and practice with domain and range.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=lj_JB8sfyIM\">Domain and Range of Square Root Functions<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=FtJRstFMdhA\">Determining Domain and Range<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=8jrkzZy04BQ\">Find Domain and Range Given the Graph<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=GPBq18fCEv4\">Find Domain and Range Given a Table<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=xOsYVyjTM0Q\">Find Domain and Range Given Points on a Coordinate Plane<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\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\">3.2 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1165135176628\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1165135172218\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1165137665109\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135245908\" 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-id1165137665109-solution\">1<\/a><span class=\"os-divider\">. <\/span>Why does the domain differ for different functions?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135440209\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135533141\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>How do we determine the domain of a function defined by an equation?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137635386\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135390940\" 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-id1165137635386-solution\">3<\/a><span class=\"os-divider\">. <\/span>Explain why the domain of\u00a0[latex]f(x)=\\sqrt[3]{x}[\/latex] is different from the domain of [latex]f(x)=\\sqrt{x}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134042454\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134042457\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134211324\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137446310\" 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-id1165134211324-solution\">5<\/a><span class=\"os-divider\">. <\/span>How do you graph a piecewise function?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137771069\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id1165137408926\">For the following exercises, find the domain of each function using interval notation.<\/p>\n<div id=\"fs-id1165137833819\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137833821\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=-2x(x-1)(x-2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137854912\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134312130\" 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-id1165137854912-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=5-2x^2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135512534\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135512537\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=3\\sqrt{x-2}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137473385\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137473388\" 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-id1165137473385-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=3-\\sqrt{6-2x}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135192268\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135192270\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\sqrt{4-3x}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137629066\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137483196\" 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-id1165137629066-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\sqrt{x^2+4}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137807107\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137551129\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\sqrt[3]{1-2x}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134259277\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134259279\" 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-id1165134259277-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\sqrt[3]{x-1}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135503751\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134156030\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex]f(x0=\\frac{9}{x-6}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133276237\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133276240\" 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-id1165133276237-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{3x+1}{4x+2}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137810520\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137810522\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{\\sqrt{x+4}}{x-4}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135548992\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135634123\" 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-id1165135548992-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{x-3}{x^2+9x-22}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135593402\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135593404\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{1}{x^2-x-6}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135191342\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134284474\" 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-id1165135191342-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{2x^3-250}{x^2-2x-15}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137921795\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137921797\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{5}{\\sqrt{x-3}}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137476914\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137476916\" 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-id1165137476914-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{2x+1}{\\sqrt{5-x}}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135185292\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137640755\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{\\sqrt{x-4}}{\\sqrt{x-6}}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135252252\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137611840\" 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-id1165135252252-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{\\sqrt{x-6}}{\\sqrt{x-4}}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137601712\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137601714\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{x}{x}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137628472\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137651574\" 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-id1165137628472-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{x^2-9x}{x^2-81}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137469452\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137469454\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span>Find the domain of the function\u00a0[latex]f(x)=\\sqrt{2x^3-50x}[\/latex] by:<\/p>\n<p>(a) using algebra.<\/p>\n<p>(b) graphing the function in the radicand and determining intervals on the <em data-effect=\"italics\">x<\/em>-axis for which the radicand is nonnegative.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137580833\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphical<\/h3>\n<p id=\"fs-id1165135186809\">For the following exercises, write the domain and range of each function using interval notation.<\/p>\n<div id=\"fs-id1165135168172\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137647479\" 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-id1165135168172-solution\">27<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165137891294\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from (2, 8].\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1185\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.27-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.27-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.27-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.27-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.27-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.27.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135160181\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135160183\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165137837830\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from [4, 8).\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1186\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.28-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.28-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.28-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.28-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.28-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.28.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137723404\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137809982\" 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-id1165137723404-solution\">29<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165137733767\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from [-4, 4].\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1187\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.29-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.29-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.29-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.29-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.29-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.29.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137590678\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134168421\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165137837060\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from [2, 6].\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1188\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.30-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.30-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.30-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.30-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.30-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.30.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137737326\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137737328\" 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-id1165137737326-solution\">31<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165134129572\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from [-5, 3).\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1189\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.31-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.31-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.31-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.31-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.31-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.31.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137404973\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137404975\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165134305418\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from [-3, 2).\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1190\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.32-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.32-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.32-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.32-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.32-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.32.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137544188\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137437269\" 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-id1165137544188-solution\">33<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165137447903\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from (-infinity, 2].\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1191\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.33-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.33-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.33-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.33-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.33-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.33.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135176309\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134323791\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165135192955\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from [-4, infinity).\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1192\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.34-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.34-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.34-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.34-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.34-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.34.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137642580\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137642582\" 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-id1165137642580-solution\">35<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165134482733\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from [-6, -1\/6]U[1\/6, 6]\/.\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1193\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.35-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.35-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.35-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.35-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.35-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.35.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137442385\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137812572\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165137645308\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from (-2.5, infinity).\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1194\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.36-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.36-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.36-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.36-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.36-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.36.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137851981\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137851983\" 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-id1165137851981-solution\">37<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165137602824\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a function from [-3, infinity).\" data-display=\"block\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1195\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.37-290x300.webp\" alt=\"\" width=\"290\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.37-290x300.webp 290w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.37-65x67.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.37-225x232.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.37-350x362.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/3.2.37.webp 362w\" sizes=\"auto, (max-width: 290px) 100vw, 290px\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137785119\">For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.<\/p>\n<div id=\"fs-id1165137462167\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137408525\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex]f(x) = \\begin{cases}x+1 & \\text{if} & x<-2 \\\\ -2x-3 & \\text{if} & x\\ge-2 \\end{cases}[\/latex]\n\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137562309\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134328320\" 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-id1165137562309-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex]f(x) = \\begin{cases}2x-1 &\u00a0 \\text{if} & x<1 \\\\ 1+x & \\text{if} & x\\ge1 \\end{cases}[\/latex]\n\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137628033\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137658060\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex]f(x) = \\begin{cases}x+1 & \\text{if} & x<0 \\\\ x-1 & \\text{if} & x>0 \\end{cases}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135641679\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135641681\" 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-id1165135641679-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex]f(x) = \\begin{cases}3 & \\text{if} & x<0 \\\\ \\sqrt{x} & \\text{if} & x\\ge0 \\end{cases}[\/latex]\n\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135192719\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135192721\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex]f(x) = \\begin{cases}x^2 & \\text{if} & x<0 \\\\ 1-x & \\text{if} & x>0 \\end{cases}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137594981\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135210029\" 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-id1165137594981-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex]f(x) = \\begin{cases}x^2 & \\text{if} & x<0 \\\\ x+2 & \\text{if} & x\\ge0 \\end{cases}[\/latex]\n\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137571389\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137433000\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex]f(x) = \\begin{cases}x+1 & \\text{if} & x<1 \\\\ x^3 & \\text{if} & x\\ge0 \\end{cases}[\/latex]\n\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137407891\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137554125\" 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-id1165137407891-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex]f(x) = \\begin{cases}|x| & \\text{if} & x<2 \\\\ 1 & \\text{if} & x\\ge2 \\end{cases}[\/latex]\n\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165134118450\" data-depth=\"2\">\n<h3 data-type=\"title\">Numeric<\/h3>\n<p id=\"fs-id1165135188383\">For the following exercises, given each function\u00a0[latex]f[\/latex] evaluate\u00a0[latex]f(-3), f(-2), f(-1),[\/latex] and [latex]f(0).[\/latex]<\/p>\n<div id=\"fs-id1165137471865\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137471867\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex]f(x) = \\begin{cases}x+1 & \\text{if} & x<-2 \\\\ -2x-3 & \\text{if} & x\\ge-2 \\end{cases}[\/latex]\n\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134122954\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134122956\" 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-id1165134122954-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex]f(x) = \\begin{cases}1 & \\text{if} & x\\le-3 \\\\ 0 & \\text{if} & x>-3 \\end{cases}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137556768\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137423742\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex]f(x) = \\begin{cases}-2x^2+3 & \\text{if} & x\\le-1 \\\\ 5x-7 & \\text{if} & x>-1 \\end{cases}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137469026\">For the following exercises, given each function\u00a0[latex]f[\/latex] evaluate\u00a0[latex]f(-1), f(0), f(2),[\/latex] and [latex]f(4).[\/latex]<\/p>\n<div id=\"fs-id1165134380351\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134380353\" 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-id1165134380351-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex]f(x) = \\begin{cases}7x+3 & \\text{if} & x<0 \\\\ 7x+6 & \\text{if} & x\\ge0 \\end{cases}[\/latex]\n\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137693713\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137679373\" data-type=\"problem\">\n<p><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex]f(x) = \\begin{cases}x^2-2 & \\text{if} & x<2 \\\\ 4+|x-5| & \\text{if} & x\\ge2 \\end{cases}[\/latex]\n\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137715004\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137715006\" 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-id1165137715004-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex]f(x) = \\begin{cases}5x & \\text{if} & x<0 \\\\ 3 & \\text{if} & x\\le0\\le3 \\\\ x^2 & \\text{if} & x>3 \\end{cases}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137837869\">For the following exercises, write the domain for the piecewise function in interval notation.<\/p>\n<div id=\"fs-id1165137837872\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135341427\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex]f(x) = \\begin{cases}x+1 & \\text{if} & x<-2 \\\\ -2x-3 & \\text{if} & x\\ge-2\\end{cases}[\/latex]\n\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137704661\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137704664\" 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-id1165137704661-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex]f(x) = \\begin{cases}x^2-2 & \\text{if} & x<1 \\\\ -x^2+2 & \\text{if} & x>1 \\end{cases}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137772429\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137772431\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span> [latex]f(x) = \\begin{cases}2x-3 & \\text{if} & x<0 \\\\ -3x^2 & \\text{if} & x\\ge2 \\end{cases}[\/latex]\n\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135194497\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<div id=\"fs-id1165137780865\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137780867\" 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-id1165137780865-solution\">55<\/a><span class=\"os-divider\">. <\/span>Graph\u00a0[latex]y=\\frac{1}{x^2}[\/latex] on the viewing window\u00a0[latex][-0.5, -0.1][\/latex] and\u00a0[latex][0.1, 0.5].[\/latex] Determine the corresponding range for the viewing window. Show the graphs.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165131911953\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137842479\" data-type=\"problem\">\n<p><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>Graph\u00a0[latex]y=\\frac{1}{x}[\/latex] on the viewing window\u00a0[latex][-0.5, -0.1][\/latex] and\u00a0[latex][0.1, 0.5].[\/latex] Determine the corresponding range for the viewing window. Show the graphs.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137733672\" data-depth=\"2\">\n<h3 data-type=\"title\">Extension<\/h3>\n<div id=\"fs-id1165137442197\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133221851\" 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-id1165137442197-solution\">57<\/a><span class=\"os-divider\">. <\/span>Suppose the range of a function\u00a0[latex]f[\/latex] is\u00a0[latex][-5, 8].[\/latex] What is the range of [latex]|f(x)|?[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137679047\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137679049\" data-type=\"problem\">\n<p><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>Create a function in which the range is all nonnegative real numbers.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135209378\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135209380\" 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-id1165135209378-solution\">59<\/a><span class=\"os-divider\">. <\/span>Create a function in which the domain is [latex]x>2.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137832031\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<div id=\"fs-id1165135511303\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135511305\" data-type=\"problem\">\n<p><span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span>A physics student at CCA is doing an experiment which tracks the motions of a rock in the air. The height\u00a0[latex]h[\/latex] of the rock is a function of the time\u00a0[latex]t[\/latex] it is in the air. The height in feet for\u00a0[latex]t[\/latex] seconds is given by the function\u00a0[latex]h(t)=-16t^2+96t.[\/latex] What is the domain of the function? What does the domain mean in the context of the problem?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137406705\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137406708\" 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-id1165137406705-solution\">61<\/a><span class=\"os-divider\">. <\/span>The cost in dollars of making\u00a0[latex]x[\/latex] items is given by the function [latex]C(x)=10x+500.[\/latex]<\/p>\n<p>(a) The fixed cost is determined when zero items are produced. Find the fixed cost for this item.<\/p>\n<p>(b) What is the cost of making 25 items?<\/p>\n<p>(c) Suppose the maximum cost allowed is $1500. What are the domain and range of the cost function, [latex]C(x)?[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<div data-type=\"footnote-refs\"><\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-133-1\">The Numbers: Where Data and the Movie Business Meet. \u201cBox Office History for Horror Movies.\u201d http:\/\/www.the-numbers.com\/market\/genre\/Horror. Accessed 3\/24\/2014 <a href=\"#return-footnote-133-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":158,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-133","chapter","type-chapter","status-publish","hentry"],"part":105,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/133","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":12,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/133\/revisions"}],"predecessor-version":[{"id":1487,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/133\/revisions\/1487"}],"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\/133\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=133"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=133"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=133"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=133"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}