Ch. 3 Key Terms – College Algebra 2e | OpenStax
Key Terms
- absolute maximum
- the greatest value of a function over an interval
- absolute minimum
- the lowest value of a function over an interval
- average rate of change
- the difference in the output values of a function found for two values of the input divided by the difference between the inputs
- composite function
- the new function formed by function composition, when the output of one function is used as the input of another
- decreasing function
- a function is decreasing in some open interval if [latex]f(b)< f(a)[/latex] for any two input values [latex]a[/latex] and [latex]b[/latex] in the given interval where [latex]b>a[/latex]
- dependent variable
- an output variable
- domain
- the set of all possible input values for a relation
- even function
- a function whose graph is unchanged by horizontal reflection, [latex]f(x)=f(-x),[/latex] and is symmetric about the y-axis
- function
- a relation in which each input value yields a unique output value
- horizontal compression
- a transformation that compresses a function’s graph horizontally, by multiplying the input by a constant [latex]b>1[/latex]
- horizontal line test
- a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once
- horizontal reflection
- a transformation that reflects a function’s graph across the y-axis by multiplying the input by [latex]-1[/latex]
- horizontal shift
- a transformation that shifts a function’s graph left or right by adding a positive or negative constant to the input
- horizontal stretch
- a transformation that stretches a function’s graph horizontally by multiplying the input by a constant [latex]0< b< 1[/latex]
- increasing function
- a function is increasing in some open interval if [latex]f(b)>f(a)[/latex] for any two input values [latex]a[/latex] and [latex]b[/latex] in the given interval where [latex]b>a[/latex]
- independent variable
- an input variable
- input
- each object or value in a domain that relates to another object or value by a relationship known as a function
- interval notation
- a method of describing a set that includes all numbers between a lower limit and an upper limit; the lower and upper values are listed between brackets or parentheses, a square bracket indicating inclusion in the set, and a parenthesis indicating exclusion
- inverse function
- for any one-to-one function [latex]f(x),[/latex] the inverse is a function [latex]- f^{-1}(x)[/latex] such that [latex]f^{-1}(f(x))=x[/latex] for all [latex]x[/latex] in the domain of [latex]f;[/latex] this also implies that [latex]f(f^{-1}(x))=x[/latex] for all [latex]x[/latex] in the domain of [latex]f^{-1}[/latex]
- local extrema
- collectively, all of a function’s local maxima and minima
- local maximum
- a value of the input where a function changes from increasing to decreasing as the input value increases.
- local minimum
- a value of the input where a function changes from decreasing to increasing as the input value increases.
- odd function
- a function whose graph is unchanged by combined horizontal and vertical reflection, [latex]f(x)=-f(-x),[/latex] and is symmetric about the origin
- one-to-one function
- a function for which each value of the output is associated with a unique input value
- output
- each object or value in the range that is produced when an input value is entered into a function
- piecewise function
- a function in which more than one formula is used to define the output
- range
- the set of output values that result from the input values in a relation
- rate of change
- the change of an output quantity relative to the change of the input quantity
- relation
- a set of ordered pairs
- set-builder notation
- a method of describing a set by a rule that all of its members obey; it takes the form [latex]\{x|\hspace{0.5em}\text{statement about}\hspace{0.5em}x\}[/latex]
- vertical compression
- a function transformation that compresses the function’s graph vertically by multiplying the output by a constant [latex]0< a< 1[/latex]
- vertical line test
- a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once
- vertical reflection
- a transformation that reflects a function’s graph across the x-axis by multiplying the output by [latex]-1[/latex]
- vertical shift
- a transformation that shifts a function’s graph up or down by adding a positive or negative constant to the output
- vertical stretch
- a transformation that stretches a function’s graph vertically by multiplying the output by a constant [latex]a>1[/latex]
