{"id":161,"date":"2023-09-20T22:47:57","date_gmt":"2023-09-20T22:47:57","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/inverse-functions\/"},"modified":"2025-08-17T16:01:49","modified_gmt":"2025-08-17T16:01:49","slug":"inverse-functions-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/inverse-functions-learn-it-1\/","title":{"raw":"Inverse Functions: Learn It 1","rendered":"Inverse Functions: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Check if a function can have an inverse by using the horizontal line test<\/li>\r\n\t<li>Determine a function\u2019s inverse and draw its mirrored graph<\/li>\r\n\t<li>Calculate values using inverse trig functions like arcsine, arccosine, and arctangent<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Inverse Functions<\/h2>\r\n<p>An inverse function reverses the operation done by a particular function. In other words, whatever a function does, the inverse function undoes it. If we have a function [latex]f[\/latex] that takes an input [latex]x[\/latex] and produces an output [latex]f(x)[\/latex], the inverse function, denoted as [latex]f^{\u22121}[\/latex], takes [latex]f(x)[\/latex] as its input and returns the original [latex]x[\/latex] as its output.\u00a0<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>inverse functions<\/h3>\r\n\r\nAn <strong>inverse function<\/strong> reverses the effect of the original function, effectively 'undoing' what the original function does. <br \/>\r\n<br \/>\r\nThe inverse of a function [latex]f[\/latex] is denoted by [latex]f^{\u22121}[\/latex].<\/div>\r\n<\/section>\r\n<section class=\"textbox proTip\">\r\n<p>Note that [latex]f^{-1}[\/latex] is read as \u201cf inverse.\u201d Here, the [latex] -1[\/latex] is not used as an exponent and [latex]f^{-1}(x) \\ne \\frac{1}{f(x)}[\/latex].<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>For instance, let's consider the function [latex]f(x)=x^3+4[\/latex]. To find the inverse, we set [latex]y= f(x)[\/latex], which gives us [latex]y=x^3+4[\/latex]. We get the inverse function [latex]f^{\u22121}(y)[\/latex] by solving for [latex]x[\/latex].<br \/>\r\n<br \/>\r\nIn this instance, subtract [latex]4[\/latex] from both sides to get [latex]t-4=x^3[\/latex], and then take the cube root of both sides. This gives us [latex]x=\\sqrt[3]{y-4}[\/latex].\u00a0 This equation defines [latex]x[\/latex] as a function of [latex]y[\/latex].<br \/>\r\n<br \/>\r\nDenoting this function as [latex]f^{-1}[\/latex], and writing [latex]x=f^{-1}(y)=\\sqrt[3]{y-4}[\/latex], we see that for any [latex]x[\/latex] in the domain of [latex]f, \\, f^{-1}(f(x))=f^{-1}(x^3+4)=x[\/latex].<\/p>\r\n<p>Thus, this new function, [latex]f^{-1}[\/latex], \u201cundid\u201d what the original function [latex]f[\/latex] did.\u00a0<\/p>\r\n<\/section>\r\n<p>Two functions are inverses of each other if the domain [latex]D[\/latex] of [latex]f[\/latex] becomes the range [latex]R[\/latex] of [latex]f^{\u22121}[\/latex] and vice versa. <br \/>\r\n<br \/>\r\nIn other words, for a function [latex]f[\/latex] and its inverse [latex]f^{-1}[\/latex],<\/p>\r\n<div id=\"fs-id1170572141883\" class=\"equation\" style=\"text-align: center;\">[latex]f^{-1}(f(x))=x[\/latex] for all [latex]x[\/latex] in [latex]D[\/latex], and [latex]f(f^{-1}(y))=y[\/latex] for all [latex]y[\/latex] in [latex]R[\/latex]<\/div>\r\n<p><br \/>\r\nFigure 1 shows the relationship between the domain and range of [latex]f[\/latex] and the domain and range of [latex]f^{-1}[\/latex].<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202523\/CNX_Calc_Figure_01_04_001.jpg\" alt=\"An image of two bubbles. The first bubble is orange and has two labels: the top label is \u201cDomain of f\u201d and the bottom label is \u201cRange of f inverse\u201d. Within this bubble is the variable \u201cx\u201d. An orange arrow with the label \u201cf\u201d points from this bubble to the second bubble. The second bubble is blue and has two labels: the top label is \u201crange of f\u201d and the bottom label is \u201cdomain of f inverse\u201d. Within this bubble is the variable \u201cy\u201d. A blue arrow with the label \u201cf inverse\u201d points from this bubble to the first bubble.\" width=\"487\" height=\"160\" \/> Figure 1. Given a function [latex]f[\/latex] and its inverse [latex]f^{-1}, \\, f^{-1}(y)=x[\/latex] if and only if [latex]f(x)=y[\/latex]. The range of [latex]f[\/latex] becomes the domain of [latex]f^{-1}[\/latex] and the domain of [latex]f[\/latex] becomes the range of [latex]f^{-1}[\/latex].[\/caption]\r\n\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]4062[\/ohm_question]<\/p>\r\n<\/section>\r\n<h3>Horizontal Line Test<\/h3>\r\n<p>Not all functions have inverses that are also functions, because for a function to have an inverse, each output must be linked to one and only one input. This unique pairing ensures that the inverse process can always match an output back to one specific input, fulfilling the definition of a function.<\/p>\r\n<p>For example, let\u2019s try to find the inverse function for [latex]f(x)=x^2[\/latex]. Solving the equation [latex]y=x^2[\/latex] for [latex]x[\/latex], we arrive at the equation [latex]x= \\pm \\sqrt{y}[\/latex]. This equation does not describe [latex]x[\/latex] as a function of [latex]y[\/latex] because there are two solutions to this equation for every [latex]y&amp;gt;0[\/latex]. <br \/>\r\n<br \/>\r\nThe problem with trying to find an inverse function for [latex]f(x)=x^2[\/latex] is that two inputs are sent to the same output for each output [latex]y&amp;gt;0[\/latex]. The function [latex]f(x)=x^3+4[\/latex] discussed earlier did not have this problem. For that function, each input was sent to a different output. <br \/>\r\n<br \/>\r\nA function that sends each input to a different output is called a <strong>one-to-one<\/strong> function.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3><strong>one-to-one<\/strong><\/h3>\r\n<p>A one-to-one function is a type of function in which each output value is paired with a unique input value.\u00a0<\/p>\r\n<\/section>\r\n<p>One way to determine whether a function is one-to-one is by looking at its graph. If a function is one-to-one, then no two inputs can be sent to the same output. Therefore, if we draw a horizontal line anywhere in the [latex]xy[\/latex]-plane, it cannot intersect the graph more than once. This is known as the <strong>horizontal line test.\u00a0<\/strong><\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>Horizontal Line Test<\/h3>\r\n<p>A function [latex]f[\/latex] is one-to-one, if and only if, every horizontal line intersects the graph of [latex]f[\/latex] no more than once.<\/p>\r\n<\/section>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202527\/CNX_Calc_Figure_01_04_002.jpg\" alt=\"An image of two graphs. Both graphs have an x axis that runs from -3 to 3 and a y axis that runs from -3 to 4. The first graph is of the function \u201cf(x) = x squared\u201d, which is a parabola. The function decreases until it hits the origin, where it begins to increase. The x intercept and y intercept are both at the origin. There are two orange horizontal lines also plotted on the graph, both of which run through the function at two points each. The second graph is of the function \u201cf(x) = x cubed\u201d, which is an increasing curved function. The x intercept and y intercept are both at the origin. There are three orange lines also plotted on the graph, each of which only intersects the function at one point.\" width=\"487\" height=\"313\" \/> Figure 3. (a) The function [latex]f(x)=x^2[\/latex] is not one-to-one because it fails the horizontal line test. (b) The function [latex]f(x)=x^3[\/latex] is one-to-one because it passes the horizontal line test.[\/caption]\r\n\r\n<section class=\"textbox proTip\">\r\n<p>Note that the horizontal line test is different from the vertical line test. The vertical line test determines whether a graph is the graph of a function. The horizontal line test determines whether a function is one-to-one (Figure 3).<\/p>\r\n<\/section>\r\n<section class=\"textbox recall\">\r\n<p><strong>The Vertical Line Test<br \/>\r\n<\/strong><br \/>\r\nThe vertical line test confirms whether a relation is a function by checking that every vertical line crosses the graph at most once.\u00a0<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"304\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232417\/image001.jpg\" alt=\"A graph of a semicircle. Four vertical lines cross the semicircle at one point each.\" width=\"304\" height=\"307\" \/> Figure 2. Semicircle graph undergoing the vertical line test.[\/caption]\r\n\r\n<p>When a vertical line is placed across the plot of this relation, it does not intersect the graph more than once for any values of [latex]x[\/latex]. This is a graph of a function.<\/p>\r\n<p>If, on the other hand, a graph shows two or more intersections with a vertical line, then an input ([latex]x[\/latex]-coordinate) can have more than one output ([latex]y[\/latex]-coordinate), and [latex]y[\/latex] is not a function of [latex]x[\/latex].\u00a0<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572150756\">For each of the following functions, use the horizontal line test to determine whether it is one-to-one.<\/p>\r\n<p>a.<\/p>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"420\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202531\/CNX_Calc_Figure_01_04_003.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 11 and the y axis runs from -3 to 11. The graph is of a step function which contains 10 horizontal steps. Each steps starts with a closed circle and ends with an open circle. The first step starts at the origin and ends at the point (1, 0). The second step starts at the point (1, 1) and ends at the point (1, 2). Each of the following 8 steps starts 1 unit higher in the y direction than where the previous step ended. The tenth and final step starts at the point (9, 9) and ends at the point (10, 9)\" width=\"420\" height=\"428\" \/> Figure 4. Graph of a step function[\/caption]\r\n\r\n<p>b.<\/p>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"420\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202534\/CNX_Calc_Figure_01_04_004.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 6 and the y axis runs from -3 to 6. The graph is of the function \u201cf(x) = (1\/x)\u201d, a curved decreasing function. The graph of the function starts right below the x axis in the 4th quadrant and begins to decreases until it comes close to the y axis. The graph keeps decreasing as it gets closer and closer to the y axis, but never touches it due to the vertical asymptote. In the first quadrant, the graph of the function starts close to the y axis and keeps decreasing until it gets close to the x axis. As the function continues to decreases it gets closer and closer to the x axis without touching it, where there is a horizontal asymptote.\" width=\"420\" height=\"438\" \/> Figure 5. Graph of f(x)[\/caption]\r\n\r\n<p>[reveal-answer q=\"fs-id1170572241370\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572241370\"]<\/p>\r\n<ol id=\"fs-id1170572241370\" style=\"list-style-type: lower-alpha;\">\r\n\t<li>Since the horizontal line [latex]y=n[\/latex] for any integer [latex]n\\ge 0[\/latex] intersects the graph more than once, this function is not one-to-one.\r\n[caption id=\"\" align=\"alignnone\" width=\"410\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202536\/CNX_Calc_Figure_01_04_005.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 11 and the y axis runs from -3 to 11. The graph is of a step function which contains 10 horizontal steps. Each steps starts with a closed circle and ends with an open circle. The first step starts at the origin and ends at the point (1, 0). The second step starts at the point (1, 1) and ends at the point (1, 2). Each of the following 8 steps starts 1 unit higher in the y direction than where the previous step ended. The tenth and final step starts at the point (9, 9) and ends at the point (10, 9). There are also two horizontal orange lines plotted on the graph, each of which run through an entire step of the function.\" width=\"410\" height=\"418\" \/> Figure 6. This is not a one-to-one function.[\/caption]\r\n<\/li>\r\n\t<li>Since every horizontal line intersects the graph once (at most), this function is one-to-one.\r\n[caption id=\"\" align=\"alignnone\" width=\"410\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202538\/CNX_Calc_Figure_01_04_006.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 6 and the y axis runs from -3 to 6. The graph is of the function \u201cf(x) = (1\/x)\u201d, a curved decreasing function. The graph of the function starts right below the x axis in the 4th quadrant and begins to decreases until it comes close to the y axis. The graph keeps decreasing as it gets closer and closer to the y axis, but never touches it due to the vertical asymptote. In the first quadrant, the graph of the function starts close to the y axis and keeps decreasing until it gets close to the x axis. As the function continues to decreases it gets closer and closer to the x axis without touching it, where there is a horizontal asymptote. There are also three horizontal orange lines plotted on the graph, each of which only runs through the function at one point.\" width=\"410\" height=\"427\" \/> Figure 7. This is a one-to-one function.[\/caption]\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]111715[\/ohm_question]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]169058[\/ohm_question]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Check if a function can have an inverse by using the horizontal line test<\/li>\n<li>Determine a function\u2019s inverse and draw its mirrored graph<\/li>\n<li>Calculate values using inverse trig functions like arcsine, arccosine, and arctangent<\/li>\n<\/ul>\n<\/section>\n<h2>Inverse Functions<\/h2>\n<p>An inverse function reverses the operation done by a particular function. In other words, whatever a function does, the inverse function undoes it. If we have a function [latex]f[\/latex] that takes an input [latex]x[\/latex] and produces an output [latex]f(x)[\/latex], the inverse function, denoted as [latex]f^{\u22121}[\/latex], takes [latex]f(x)[\/latex] as its input and returns the original [latex]x[\/latex] as its output.\u00a0<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>inverse functions<\/h3>\n<p>An <strong>inverse function<\/strong> reverses the effect of the original function, effectively &#8216;undoing&#8217; what the original function does. <\/p>\n<p>The inverse of a function [latex]f[\/latex] is denoted by [latex]f^{\u22121}[\/latex].<\/p><\/div>\n<\/section>\n<section class=\"textbox proTip\">\n<p>Note that [latex]f^{-1}[\/latex] is read as \u201cf inverse.\u201d Here, the [latex]-1[\/latex] is not used as an exponent and [latex]f^{-1}(x) \\ne \\frac{1}{f(x)}[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>For instance, let&#8217;s consider the function [latex]f(x)=x^3+4[\/latex]. To find the inverse, we set [latex]y= f(x)[\/latex], which gives us [latex]y=x^3+4[\/latex]. We get the inverse function [latex]f^{\u22121}(y)[\/latex] by solving for [latex]x[\/latex].<\/p>\n<p>In this instance, subtract [latex]4[\/latex] from both sides to get [latex]t-4=x^3[\/latex], and then take the cube root of both sides. This gives us [latex]x=\\sqrt[3]{y-4}[\/latex].\u00a0 This equation defines [latex]x[\/latex] as a function of [latex]y[\/latex].<\/p>\n<p>Denoting this function as [latex]f^{-1}[\/latex], and writing [latex]x=f^{-1}(y)=\\sqrt[3]{y-4}[\/latex], we see that for any [latex]x[\/latex] in the domain of [latex]f, \\, f^{-1}(f(x))=f^{-1}(x^3+4)=x[\/latex].<\/p>\n<p>Thus, this new function, [latex]f^{-1}[\/latex], \u201cundid\u201d what the original function [latex]f[\/latex] did.\u00a0<\/p>\n<\/section>\n<p>Two functions are inverses of each other if the domain [latex]D[\/latex] of [latex]f[\/latex] becomes the range [latex]R[\/latex] of [latex]f^{\u22121}[\/latex] and vice versa. <\/p>\n<p>In other words, for a function [latex]f[\/latex] and its inverse [latex]f^{-1}[\/latex],<\/p>\n<div id=\"fs-id1170572141883\" class=\"equation\" style=\"text-align: center;\">[latex]f^{-1}(f(x))=x[\/latex] for all [latex]x[\/latex] in [latex]D[\/latex], and [latex]f(f^{-1}(y))=y[\/latex] for all [latex]y[\/latex] in [latex]R[\/latex]<\/div>\n<p>\nFigure 1 shows the relationship between the domain and range of [latex]f[\/latex] and the domain and range of [latex]f^{-1}[\/latex].<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202523\/CNX_Calc_Figure_01_04_001.jpg\" alt=\"An image of two bubbles. The first bubble is orange and has two labels: the top label is \u201cDomain of f\u201d and the bottom label is \u201cRange of f inverse\u201d. Within this bubble is the variable \u201cx\u201d. An orange arrow with the label \u201cf\u201d points from this bubble to the second bubble. The second bubble is blue and has two labels: the top label is \u201crange of f\u201d and the bottom label is \u201cdomain of f inverse\u201d. Within this bubble is the variable \u201cy\u201d. A blue arrow with the label \u201cf inverse\u201d points from this bubble to the first bubble.\" width=\"487\" height=\"160\" \/><figcaption class=\"wp-caption-text\">Figure 1. Given a function [latex]f[\/latex] and its inverse [latex]f^{-1}, \\, f^{-1}(y)=x[\/latex] if and only if [latex]f(x)=y[\/latex]. The range of [latex]f[\/latex] becomes the domain of [latex]f^{-1}[\/latex] and the domain of [latex]f[\/latex] becomes the range of [latex]f^{-1}[\/latex].<\/figcaption><\/figure>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm4062\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=4062&theme=lumen&iframe_resize_id=ohm4062&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<h3>Horizontal Line Test<\/h3>\n<p>Not all functions have inverses that are also functions, because for a function to have an inverse, each output must be linked to one and only one input. This unique pairing ensures that the inverse process can always match an output back to one specific input, fulfilling the definition of a function.<\/p>\n<p>For example, let\u2019s try to find the inverse function for [latex]f(x)=x^2[\/latex]. Solving the equation [latex]y=x^2[\/latex] for [latex]x[\/latex], we arrive at the equation [latex]x= \\pm \\sqrt{y}[\/latex]. This equation does not describe [latex]x[\/latex] as a function of [latex]y[\/latex] because there are two solutions to this equation for every [latex]y&gt;0[\/latex]. <\/p>\n<p>The problem with trying to find an inverse function for [latex]f(x)=x^2[\/latex] is that two inputs are sent to the same output for each output [latex]y&gt;0[\/latex]. The function [latex]f(x)=x^3+4[\/latex] discussed earlier did not have this problem. For that function, each input was sent to a different output. <\/p>\n<p>A function that sends each input to a different output is called a <strong>one-to-one<\/strong> function.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3><strong>one-to-one<\/strong><\/h3>\n<p>A one-to-one function is a type of function in which each output value is paired with a unique input value.\u00a0<\/p>\n<\/section>\n<p>One way to determine whether a function is one-to-one is by looking at its graph. If a function is one-to-one, then no two inputs can be sent to the same output. Therefore, if we draw a horizontal line anywhere in the [latex]xy[\/latex]-plane, it cannot intersect the graph more than once. This is known as the <strong>horizontal line test.\u00a0<\/strong><\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>Horizontal Line Test<\/h3>\n<p>A function [latex]f[\/latex] is one-to-one, if and only if, every horizontal line intersects the graph of [latex]f[\/latex] no more than once.<\/p>\n<\/section>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202527\/CNX_Calc_Figure_01_04_002.jpg\" alt=\"An image of two graphs. Both graphs have an x axis that runs from -3 to 3 and a y axis that runs from -3 to 4. The first graph is of the function \u201cf(x) = x squared\u201d, which is a parabola. The function decreases until it hits the origin, where it begins to increase. The x intercept and y intercept are both at the origin. There are two orange horizontal lines also plotted on the graph, both of which run through the function at two points each. The second graph is of the function \u201cf(x) = x cubed\u201d, which is an increasing curved function. The x intercept and y intercept are both at the origin. There are three orange lines also plotted on the graph, each of which only intersects the function at one point.\" width=\"487\" height=\"313\" \/><figcaption class=\"wp-caption-text\">Figure 3. (a) The function [latex]f(x)=x^2[\/latex] is not one-to-one because it fails the horizontal line test. (b) The function [latex]f(x)=x^3[\/latex] is one-to-one because it passes the horizontal line test.<\/figcaption><\/figure>\n<section class=\"textbox proTip\">\n<p>Note that the horizontal line test is different from the vertical line test. The vertical line test determines whether a graph is the graph of a function. The horizontal line test determines whether a function is one-to-one (Figure 3).<\/p>\n<\/section>\n<section class=\"textbox recall\">\n<p><strong>The Vertical Line Test<br \/>\n<\/strong><br \/>\nThe vertical line test confirms whether a relation is a function by checking that every vertical line crosses the graph at most once.\u00a0<\/p>\n<figure style=\"width: 304px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/04\/18232417\/image001.jpg\" alt=\"A graph of a semicircle. Four vertical lines cross the semicircle at one point each.\" width=\"304\" height=\"307\" \/><figcaption class=\"wp-caption-text\">Figure 2. Semicircle graph undergoing the vertical line test.<\/figcaption><\/figure>\n<p>When a vertical line is placed across the plot of this relation, it does not intersect the graph more than once for any values of [latex]x[\/latex]. This is a graph of a function.<\/p>\n<p>If, on the other hand, a graph shows two or more intersections with a vertical line, then an input ([latex]x[\/latex]-coordinate) can have more than one output ([latex]y[\/latex]-coordinate), and [latex]y[\/latex] is not a function of [latex]x[\/latex].\u00a0<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572150756\">For each of the following functions, use the horizontal line test to determine whether it is one-to-one.<\/p>\n<p>a.<\/p>\n<figure style=\"width: 420px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202531\/CNX_Calc_Figure_01_04_003.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 11 and the y axis runs from -3 to 11. The graph is of a step function which contains 10 horizontal steps. Each steps starts with a closed circle and ends with an open circle. The first step starts at the origin and ends at the point (1, 0). The second step starts at the point (1, 1) and ends at the point (1, 2). Each of the following 8 steps starts 1 unit higher in the y direction than where the previous step ended. The tenth and final step starts at the point (9, 9) and ends at the point (10, 9)\" width=\"420\" height=\"428\" \/><figcaption class=\"wp-caption-text\">Figure 4. Graph of a step function<\/figcaption><\/figure>\n<p>b.<\/p>\n<figure style=\"width: 420px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202534\/CNX_Calc_Figure_01_04_004.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 6 and the y axis runs from -3 to 6. The graph is of the function \u201cf(x) = (1\/x)\u201d, a curved decreasing function. The graph of the function starts right below the x axis in the 4th quadrant and begins to decreases until it comes close to the y axis. The graph keeps decreasing as it gets closer and closer to the y axis, but never touches it due to the vertical asymptote. In the first quadrant, the graph of the function starts close to the y axis and keeps decreasing until it gets close to the x axis. As the function continues to decreases it gets closer and closer to the x axis without touching it, where there is a horizontal asymptote.\" width=\"420\" height=\"438\" \/><figcaption class=\"wp-caption-text\">Figure 5. Graph of f(x)<\/figcaption><\/figure>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572241370\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572241370\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1170572241370\" style=\"list-style-type: lower-alpha;\">\n<li>Since the horizontal line [latex]y=n[\/latex] for any integer [latex]n\\ge 0[\/latex] intersects the graph more than once, this function is not one-to-one.<br \/>\n<figure style=\"width: 410px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202536\/CNX_Calc_Figure_01_04_005.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 11 and the y axis runs from -3 to 11. The graph is of a step function which contains 10 horizontal steps. Each steps starts with a closed circle and ends with an open circle. The first step starts at the origin and ends at the point (1, 0). The second step starts at the point (1, 1) and ends at the point (1, 2). Each of the following 8 steps starts 1 unit higher in the y direction than where the previous step ended. The tenth and final step starts at the point (9, 9) and ends at the point (10, 9). There are also two horizontal orange lines plotted on the graph, each of which run through an entire step of the function.\" width=\"410\" height=\"418\" \/><figcaption class=\"wp-caption-text\">Figure 6. This is not a one-to-one function.<\/figcaption><\/figure>\n<\/li>\n<li>Since every horizontal line intersects the graph once (at most), this function is one-to-one.<br \/>\n<figure style=\"width: 410px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202538\/CNX_Calc_Figure_01_04_006.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 6 and the y axis runs from -3 to 6. The graph is of the function \u201cf(x) = (1\/x)\u201d, a curved decreasing function. The graph of the function starts right below the x axis in the 4th quadrant and begins to decreases until it comes close to the y axis. The graph keeps decreasing as it gets closer and closer to the y axis, but never touches it due to the vertical asymptote. In the first quadrant, the graph of the function starts close to the y axis and keeps decreasing until it gets close to the x axis. As the function continues to decreases it gets closer and closer to the x axis without touching it, where there is a horizontal asymptote. There are also three horizontal orange lines plotted on the graph, each of which only runs through the function at one point.\" width=\"410\" height=\"427\" \/><figcaption class=\"wp-caption-text\">Figure 7. This is a one-to-one function.<\/figcaption><\/figure>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm111715\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=111715&theme=lumen&iframe_resize_id=ohm111715&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm169058\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=169058&theme=lumen&iframe_resize_id=ohm169058&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":6,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"1.4 Inverse Functions\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":759,"module-header":"learn_it","content_attributions":[{"type":"cc","description":"Calculus Volume 1","author":"Gilbert Strang, Edwin (Jed) Herman","organization":"OpenStax","url":"https:\/\/openstax.org\/details\/books\/calculus-volume-1","project":"","license":"cc-by-nc-sa","license_terms":"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction"},{"type":"original","description":"1.4 Inverse Functions","author":"Ryan Melton","organization":"","url":"","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/161"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":30,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/161\/revisions"}],"predecessor-version":[{"id":4743,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/161\/revisions\/4743"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/759"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/161\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=161"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=161"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=161"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=161"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}