{"id":1055,"date":"2022-03-10T22:09:29","date_gmt":"2022-03-10T22:09:29","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=1055"},"modified":"2025-11-17T18:27:58","modified_gmt":"2025-11-17T18:27:58","slug":"2-5-the-inverse-of-a-linear-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/2-5-the-inverse-of-a-linear-function\/","title":{"raw":"2.5: The Inverse of a Linear Function","rendered":"2.5: The Inverse of a Linear Function"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Objectives<\/h1>\r\n<ul>\r\n \t<li>Explain that the inverse of a linear function is a function<\/li>\r\n \t<li>Graph the inverse of a linear function<\/li>\r\n \t<li>Find the inverse function of a linear function algebraically<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>The Inverse of a Function<\/h2>\r\nWe learned in section 1.3 that the inverse of a function may be obtained by exchanging the domain and range (or the input and output) of the function. For example, the inverse of the function [latex] \\{(1, 2), (3, 4), (5, 6)\\}[\/latex] is [latex]\\{(2, 1), (4, 3), (6, 5)\\}[\/latex]. All we have to do is switch the [latex]x[\/latex]- and [latex]y[\/latex]-values in the ordered pairs. Note that the inverse of a function may or may not be a function. Figure 1 shows a function and its inverse where the inverse is a function. Figure 2 shows a function and its inverse where the inverse is NOT a function. A function will have an inverse function only if the original function is one-to-one. If the inverse of a function [latex]f[\/latex] is also a function, we may use the notation [latex]f^{-1}[\/latex] to name the inverse function. It is important to note that the \"\u20131\" is not an exponent. It is a superscript meaning \"inverse\". The notation [latex]f^{-1}[\/latex] means the inverse function of the function [latex]f[\/latex]. This function [latex]f^{-1}[\/latex] is a completely different function from [latex]f[\/latex].\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%;\">\r\n<div class=\"mceTemp\">True 1 - 1 function and its inverse<\/div><\/th>\r\n<th style=\"width: 50%;\">\r\n<div class=\"mceTemp\">Not a 1 - 1 function<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_807\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-807 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/18221231\/tan-inverse-300x300.png\" alt=\"inverse functions\" width=\"300\" height=\"300\" \/> Figure 1. The inverse of a 1-1 function is a 1-1 function.[\/caption]<\/td>\r\n<td style=\"width: 50%;\">\r\n\r\n[caption id=\"attachment_1390\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1390 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07010603\/inverse-of-fxx%5E2-300x300.png\" alt=\"Graph of y=x^2 and its inverse\" width=\"300\" height=\"300\" \/> Figure 2. A function that is not 1-1 has an inverse that is not a function.[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>The Inverse of a Linear Function<\/h2>\r\nA linear function [latex]f(x)=mx+b[\/latex] with [latex]m\\ne0[\/latex] is a one-to-one mapping of [latex]x[\/latex] to the function value [latex]f(x)[\/latex] because each point on the graph of the function has a unique input and a unique output. There are no two points that have the same input ([latex]x[\/latex]-coordinate) or the same output ([latex]y[\/latex]-coordinate). Figure 3 illustrates some points (or solution pairs) on the graph of the function [latex]f(x)=2x[\/latex] as well as the corresponding points on the graph of the inverse function [latex]f^{-1}(x)[\/latex], which are found by simply switching the [latex]x[\/latex]- and [latex]y[\/latex]-coordinates of the original points. Both sets of points show a one-to-one mapping. This is an example of the fact that the inverse of a linear function with a non-zero slope is also a one-to-one linear function.\r\n\r\n[caption id=\"attachment_801\" align=\"aligncenter\" width=\"298\"]<img class=\"wp-image-801 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-points-298x300.png\" alt=\"One function of black points (1,2), (2,4), (3,6), (4,8), and (5,10), and the inverse function (2,1), (4,2), (6,3), (8,4), and (10,5).\" width=\"298\" height=\"300\" \/> Figure 3. Points (solution pairs) on the graph of the function [latex]f(x) = 2x[\/latex] and its inverse.[\/caption]A linear function with a zero slope (a horizontal line), i.e., [latex]f(x)=b[\/latex], has an inverse that is not a function, [latex]x=b[\/latex] (a vertical line). A horizontal line does not represent a one-to-one function, which is why its inverse is not a function (figure 4).\r\n\r\n[caption id=\"attachment_1942\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1942 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/02202149\/desmos-graph-2022-05-02T142129.550-300x300.png\" alt=\"Horizontal and vertical lines\" width=\"300\" height=\"300\" \/> Figure 4. Horizontal line is a function; vertical line is not a function.[\/caption]\r\n<h2>Graphing the Inverse of a Linear Function<\/h2>\r\nTo graph the inverse of a linear function, we may start by finding two or more points (solution pairs) on the graph of the linear function. Then we simply switch the [latex]x[\/latex]- and\u00a0[latex]y[\/latex]-coordinates of each point to find points that lie on the graph of the inverse function. We can then use these points to graph the inverse function. For example, (0, 0) and (1, 2) are two points that lie on the graph of the function [latex]f(x)=2x[\/latex]. To graph the inverse function, we switch the [latex]x[\/latex]- and [latex]y[\/latex]-coordinates of these points to get (0, 0) and (2, 1). Therefore, the graph of the inverse function will be the line that passes through the two points (0, 0) and (2, 1) (See Figure 5).\r\n\r\n[caption id=\"attachment_1068\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1068 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2-5-1-InverseGraph-2-300x300.png\" alt=\"A graph with a line through (0,0) and (2,1), with it's inverse, a line through (0,0) and (1,2).\" width=\"300\" height=\"300\" \/> Figure 5. Graph of the function [latex]f(x)=2x[\/latex] and its inverse.[\/caption]\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nUse the graphs of the linear functions to graph their inverse function.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%;\">Problem 1<\/th>\r\n<th style=\"width: 50%;\">Problem 2<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">&nbsp;\r\n\r\n[caption id=\"attachment_1392\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1392\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07014041\/fx4x-2-300x298.png\" alt=\"Graph of f(x)=4x-2\" width=\"300\" height=\"298\" \/> [latex]f(x)=4x-2[\/latex][\/caption]<\/td>\r\n<td style=\"width: 50%;\">&nbsp;\r\n\r\n[caption id=\"attachment_1395\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1395 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07015330\/fx-23x5-300x300.png\" alt=\"Graph of f(x)=-2\/3 x+5\" width=\"300\" height=\"300\" \/> [latex]f(x)=-\\frac{2}{3}x+5[\/latex][\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Solution<\/h4>\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%;\">Problem 1<\/th>\r\n<th style=\"width: 50%;\">\r\n<p style=\"text-align: initial;\">Problem 2<\/p>\r\n<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">&nbsp;\r\n<p style=\"text-align: initial;\">Choose 2 (or more) points on the original line: (0, \u20132) and (1, 2)<\/p>\r\n<p style=\"text-align: initial;\">Reverse the [latex]x[\/latex]- and [latex]y[\/latex]-coordinates: (\u20132, 0) and (2, 1)<\/p>\r\n<p style=\"text-align: initial;\">Plot the new points. The line that passes through them is the inverse function.<\/p>\r\n\r\n[caption id=\"attachment_1391\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1391\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07013646\/fx4x-2-and-inverse-300x294.png\" alt=\"Graph of f(x)=4x-2 and its inverse\" width=\"300\" height=\"294\" \/> [latex]f(x)=4x-2[\/latex] and its inverse[\/caption]\r\n<p style=\"text-align: initial;\"><\/p>\r\n<\/td>\r\n<td style=\"width: 50%;\">\r\n<p style=\"text-align: initial;\">Choose 2 (or more) points on the original line: (0, 5) and (6, 1)<\/p>\r\n<p style=\"text-align: initial;\">Reverse the [latex]x[\/latex]- and [latex]y[\/latex]-coordinates: (5, 0) and (1, 6)<\/p>\r\n<p style=\"text-align: initial;\">Plot the new points. The line that passes through them is the inverse function.<\/p>\r\n\r\n[caption id=\"attachment_1396\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1396\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07015336\/fx-23x5-and-inverse-300x296.png\" alt=\"Graph of f(x)=-2\/3 x + 5 and its inverse\" width=\"300\" height=\"296\" \/> [latex]f(x)=-\\frac{2}{3}x+5[\/latex] and its inverse[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nUse the graphs of the linear functions to graph their inverse function.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">Problem 1<\/td>\r\n<td style=\"width: 50%;\">Problem 2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">&nbsp;\r\n\r\n[caption id=\"attachment_1393\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1393\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07014325\/fx-3x5-300x297.png\" alt=\"Graph of f(x)=-3x+5\" width=\"300\" height=\"297\" \/> [latex]f(x)=-3x+5[\/latex][\/caption]<\/td>\r\n<td style=\"width: 50%;\">&nbsp;\r\n\r\n[caption id=\"attachment_1397\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1397\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07020047\/fx43x2-300x294.png\" alt=\"Graph of f(x)=4\/3 x+2\" width=\"300\" height=\"294\" \/> [latex]f(x)=\\frac{4}{3}x+2[\/latex][\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"hjm853\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm853\"]\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">Problem 1<\/td>\r\n<td style=\"width: 50%;\">Problem 2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">&nbsp;\r\n\r\n[caption id=\"attachment_1394\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1394\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07014620\/fx-3x5-and-inverse-300x300.png\" alt=\"Graph of f(x)=-3x+5 and its inverse\" width=\"300\" height=\"300\" \/> [latex]f(x)=-3x+5[\/latex] and its inverse[\/caption]<\/td>\r\n<td style=\"width: 50%;\">&nbsp;\r\n\r\n[caption id=\"attachment_1398\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1398\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07020052\/fx43x2-and-inverse-300x297.png\" alt=\"Graph of f(x)=4\/3 x+2 and its inverse\" width=\"300\" height=\"297\" \/> [latex]f(x)=\\frac{4}{3}x+2[\/latex] and its inverse[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nGraph the function [latex]f(x)=\\frac{2}{5}x+2[\/latex], then graph its inverse.\r\n<h4>Solution<\/h4>\r\nTo graph\u00a0[latex]f(x)=\\frac{2}{5}x+2[\/latex] we can either use a table of values or use the slope and [latex]y[\/latex]-intercept.\r\n\r\nThe slope of the line is [latex]m=\\frac{2}{5}[\/latex] and the [latex]y[\/latex]-intercept is (0, 2).\r\n\r\nTo graph the line, we start at (0, 2) then run 5 units to the right and 2 units up to get to (5, 4).\r\n\r\n[caption id=\"attachment_1400\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1400\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07021055\/fx25-x2-300x291.png\" alt=\"Graph of f(x)=2\/5 x+2 showing points at (0,2) and (5,4).\" width=\"300\" height=\"291\" \/> [latex]f(x)=\\frac{2}{5}x+2[\/latex][\/caption]Now reverse the points (0, 2) and (5, 4) to (2, 0) and (4, 5) and draw the line of the inverse function through these new points.[caption id=\"attachment_1399\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1399\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07021050\/fx25-x2-and-inverse-300x297.png\" alt=\"Graph of f(x)=2\/5 x+2 showing points (0,2) and (5,4), and its inverse showing points (2,0) and (4,5).\" width=\"300\" height=\"297\" \/> [latex]f(x)=\\frac{2}{5}x+2[\/latex] and its inverse[\/caption]<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nGraph the function [latex]f(x)=\\frac{1}{3}x-4[\/latex], then graph its inverse.\r\n\r\n[reveal-answer q=\"hjm277\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm277\"]\r\n\r\n[caption id=\"attachment_1401\" align=\"aligncenter\" width=\"362\"]<img class=\"wp-image-1401\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07021926\/fx13-x-4-and-inverse-300x292.png\" alt=\"Graph of f(x)=1\/3 x-4 and inverse. Slopes described below.\" width=\"362\" height=\"352\" \/> [latex]f(x)=\\frac{1}{3}x-4[\/latex] and its inverse[\/caption]Notice the relationship of the slope of the original function to the slope of the inverse function. The original function has a slope of 3, while the inverse function has a slope of [latex]\\frac{1}{3}[\/latex]. i.e. the slopes are reciprocals of each other, but NOT negative reciprocals like perpendicular lines.[\/hidden-answer]<\/div>\r\n<h2>Inverse Functions and Symmetry<\/h2>\r\nAs we saw in chapter 1, the graph of any function and its inverse are symmetric across the line [latex]y=x[\/latex].\u00a0 For example, figure 6 shows the graph of [latex]f(x)=\\frac{2}{5}x+2[\/latex] and its inverse. With the added line [latex]y=x[\/latex], we can see that the two lines are symmetric (mirror images of one another) across the line [latex]y=x[\/latex].\r\n\r\n<img class=\"wp-image-1409 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07033342\/Inverse-functions-and-symmetry-300x293.png\" alt=\"Graph showing symmetry of inverse functions across the line y=x\" width=\"318\" height=\"310\" \/>\r\n<p style=\"text-align: center;\">Figure 6. [latex]f(x)=\\frac{2}{5}x+2[\/latex] and its inverse<\/p>\r\nNotice the [latex]x[\/latex]- and [latex]y[\/latex]-intercepts. The [latex]y[\/latex]-intercept of (0, 2) in the original function (blue line) reflects to the [latex]x[\/latex]-intercept (2, 0) in the inverse function (green line). Also, the\u00a0[latex]x[\/latex]-intercept of (\u20135, 0) in the original function (blue line) reflects to the [latex]y[\/latex]-intercept (0, \u20135) in the inverse function (green line).\r\n\r\nThe slopes of each function are also related. The function [latex]f(x)[\/latex] has a slope of [latex]\\frac{2}{5}[\/latex], while the inverse function [latex]f^{-1}(x)[\/latex] has a slope of [latex]\\frac{5}{2}[\/latex]. <strong>The slopes of inverse functions are reciprocals of each other. <\/strong>This is\u00a0because the slope of a function is [latex]\\dfrac{\\text{change in y}}{\\text{change in x}}[\/latex]. The slope of the inverse function becomes\u00a0[latex]\\dfrac{\\text{change in x}}{\\text{change in y}}[\/latex].\r\n\r\nSince we know the slopes are reciprocals and the [latex]y[\/latex]-intercept of the inverse function is the flipped coordinates of\u00a0 the [latex]x[\/latex]-intercept of the original function, we can find the equation of the inverse function: [latex]m=\\frac{5}{2},\\;b=-5[\/latex] so [latex]f^{-1}(x)=\\frac{5}{2}x-5[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nGraph the function [latex]g(x)=\\frac{3}{7}x-2[\/latex] then use symmetry to graph its inverse. Write the equation of the inverse function.\r\n<h4>Solution<\/h4>\r\nTo graph\u00a0[latex]g(x)=\\frac{3}{7}x-2[\/latex], we can start at the [latex]y[\/latex]-intercept (0, \u20132) then use the slope of [latex]\\frac{3}{7}[\/latex] to run 7 and rise 3 to get to another point on the line: (7, 1). Then draw the line through these points.\r\n\r\n<img class=\"aligncenter wp-image-1436 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/gx37-x-2-plus-slope-300x292.png\" alt=\"Graph of g(x) showing an intercept of (0,-2) with the slope of rise 3 and run 7 yielding a second point of (7,1).\" width=\"300\" height=\"292\" \/>\r\n\r\nTo graph the inverse function, we can draw the line [latex]y=x[\/latex] then reflect points on the line of the original function across [latex]y=x[\/latex] to get points on the inverse function. Then we can draw the line of the inverse function.\r\n\r\n<img class=\"aligncenter wp-image-1437 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/gx37x-2-and-inverse-300x294.png\" alt=\"G(x) and its inverse. The inverse goes through (-2,0) and (1,7).\" width=\"300\" height=\"294\" \/>\r\n\r\nTo write the equation of the inverse function, we know that the slope is [latex]\\frac{7}{3}[\/latex]; the reciprocal of the slope of the original function.\r\n\r\nFrom the graph, the [latex]y[\/latex]-intercept is fractional but we know that the inverse goes through the point (\u20132, 0); the flipped point of (0 \u20132) on the original function.\r\n\r\nThe inverse function is linear, so has the form [latex]f(x)=mx+b=\\frac{7}{3}x+b[\/latex] with the substituted slope.\r\n\r\nTo find [latex]b[\/latex], we can use the point (\u20132, 0):\r\n\r\n[latex]\\begin{aligned}f(x)&amp;=\\frac{7}{3}x+b\\\\ 0&amp;=\\frac{7}{3}\\cdot (-2)+b\\\\ 0&amp;=\\frac{-14}{3}+b\\\\b&amp;= \\frac{14}{3}\\end{aligned}[\/latex]\r\n\r\nConsequently the inverse function is [latex]g^{-1}(x)=\\frac{7}{3}x+\\frac{14}{3}[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nGraph the function [latex]g(x)=\\frac{1}{2}x+3[\/latex] then use symmetry to graph its inverse. Write the equation of the inverse function.\r\n\r\n[reveal-answer q=\"hjm840\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm840\"]\r\n\r\n<img class=\"aligncenter wp-image-1438\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/08202803\/gx12-x3-and-inverse-300x296.png\" alt=\"Graph of g(x) with y intercept (0,3) and slope of 1\/2,and its inverse with x intercept of (3,0) and slope of 2.\" width=\"405\" height=\"400\" \/>\r\n<p style=\"text-align: center;\">[latex]g^{-1}(x)=2x-6[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nFigure 7 demonstrates the symmetry across the line [latex]y=x[\/latex] of a function and its inverse. Each point on the line of the original function has a reflected point across [latex]y=x[\/latex].\r\n\r\n&nbsp;\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/opmximd6d0?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/p>\r\n<p style=\"text-align: center;\">Figure 7. The symmetry of a function and its inverse.<\/p>\r\n\r\n<h2>The Algebraic Inverse Function of a Linear Function<\/h2>\r\nWe have learned that to graph an inverse function, we simply switch the [latex]x[\/latex]- and [latex]y[\/latex]-coordinates on the graph. The same is true for finding the inverse function algebraically. To find the inverse function of a given linear function, we switch the independent variable (e.g.,\u00a0[latex]x[\/latex]) to [latex]y[\/latex] so that it becomes the dependent variable, and the dependent variable (e.g.,\u00a0[latex]y=f(x)[\/latex]) to [latex]x[\/latex] to become the independent variable. The inverse of a function is the exchange of the domain and range of the function.\r\n\r\nFrom the perspective of an equation, a linear equation in the form [latex]y=mx+b[\/latex], becomes its inverse by switching [latex]x[\/latex] and [latex]y[\/latex] to get [latex]x=my+b[\/latex].\r\n\r\nFor example, if\u00a0[latex]y = 2x + 1[\/latex], the inverse is [latex]x=2y+1[\/latex].\r\n\r\nFor the function\u00a0[latex]f(x) = 2x + 1[\/latex], we can start by writing [latex]y=f(x)[\/latex] so that\u00a0[latex]y = 2x + 1[\/latex]. Then it is easier to switch the variables to find the inverse\u00a0[latex]x = 2y + 1[\/latex].\r\n\r\nTo write this inverse equation as a function, we need to solve it for [latex]y[\/latex] by first subtracting 1 from both sides, then dividing both sides by 2:\r\n<p style=\"text-align: center;\">[latex]\\begin{equation}\\begin{aligned}x &amp;= 2y+1\\\\x-1&amp;=2y\\\\\\frac{x-1}{2}&amp;=y\\\\\\frac{1}{2}x-\\frac{1}{2}&amp;=y\\end{aligned}\\end{equation}[\/latex]<\/p>\r\nThe last step for finding the inverse function is to use function notation and write [latex]f^{-1}(x)=y[\/latex]:\r\n<p style=\"text-align: center;\">[latex]f^{-1}(x) = \\frac{1}{2}x - \\frac{1}{2}[\/latex]<\/p>\r\n<span style=\"font-size: 1rem; text-align: initial;\">This method of switching [latex]x[\/latex] and [latex]y[\/latex] is based on what we learned through graphing functions and their inverses on the coordinate plane, where the independent variable is always represented by the variable [latex]x[\/latex] and the dependent variable is always represented by the variable [latex]y[\/latex]. In other words, the\u00a0[latex]x-axis[\/latex] represents the independent variable and the\u00a0[latex]y-axis[\/latex] the dependent variable. The final step of replacing\u00a0[latex]y[\/latex] with\u00a0[latex]f^{-1}(x)[\/latex] is because the inverse is a function related to the original function and although we need to name it differently from the original function [latex]f(x)[\/latex], we want to keep the relationship this inverse function has to the original function [latex]f(x)[\/latex].<\/span>\r\n\r\nRemember that in function notation the function name does not have to be [latex]f[\/latex], nor does the independent variable have to be [latex]x[\/latex]. It's only when graphing that [latex]x[\/latex] and [latex]y[\/latex] are typically used.\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nFind the inverse of the linear function:\r\n<ol>\r\n \t<li>[latex]f(x)=2x[\/latex]<\/li>\r\n \t<li>[latex]g(x)=x+6[\/latex]<\/li>\r\n \t<li>[latex]h(x)=4x+5[\/latex]<\/li>\r\n \t<li>[latex]s(t)=\\frac{3}{4}t-2[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\nStart by replacing the function name with [latex]y[\/latex]; switch the variables; solve for [latex]y[\/latex]; write inverse function notation for [latex]y[\/latex].\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n<p style=\"text-align: initial;\">Finding an inverse<\/p>\r\n<\/td>\r\n<td style=\"width: 50%;\">Four basic steps<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">\r\n<p style=\"text-align: initial;\">1.<\/p>\r\n<p style=\"text-align: initial;\">[latex]\\begin{equation}\\begin{aligned}f(x) &amp;=2x\\\\y &amp;=2x\\\\x &amp;=2y\\\\ \\frac{1}{2}x &amp;=y \\\\f^{-1}(x) &amp;=\\frac{1}{2}x\\end{aligned}\\end{equation}[\/latex]<\/p>\r\n<\/td>\r\n<td style=\"width: 50%;\">2.\r\n\r\n[latex]\\begin{equation}\\begin{aligned}g(x)&amp;=x+6\\\\y &amp;=x+6\\\\x &amp;=y+6\\\\ x-6 &amp;=y \\\\g^{-1}(x) &amp;=x-6\\end{aligned}\\end{equation}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">3.\r\n\r\n[latex]\\begin{equation}\\begin{aligned}h(x)&amp;=4x+5\\\\y &amp;=4x+5\\\\x &amp;=4y+5\\\\x-5 &amp;=4y\\\\ \\frac{x-5}{4}&amp;=y \\\\h^{-1}(x) &amp;=\\frac{1}{4}x-\\frac{5}{4}\\end{aligned}\\end{equation}[\/latex]<\/td>\r\n<td style=\"width: 50%;\">4.\r\n\r\n[latex]\\begin{equation}\\begin{aligned}s(t)&amp;=\\frac{3}{4}t-2\\\\y &amp;=\\frac{3}{4}t-2\\\\t &amp;=\\frac{3}{4}y-2\\\\ t+2 &amp;=\\frac{3}{4}y\\\\ \\frac{4}{3}(t+2)&amp;=y \\\\s^{-1}(t) &amp;=\\frac{4}{3}t+\\frac{8}{3}\\end{aligned}\\end{equation}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nFind the inverse of the linear function:\r\n<ol>\r\n \t<li>[latex]f(x)=-6x[\/latex]<\/li>\r\n \t<li>[latex]g(x)=2x-1[\/latex]<\/li>\r\n \t<li>[latex]h(x)=-3x+5[\/latex]<\/li>\r\n \t<li>[latex]s(t)=\\frac{1}{5}t-2[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm143\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm143\"]\r\n\r\n1. [latex]f^{-1}(x)=-\\frac{1}{6}x[\/latex]\r\n\r\n2.\u00a0[latex]g^{-1}(x)=\\frac{1}{2}x+\\frac{1}{2}[\/latex]\r\n\r\n3.\u00a0[latex]h^{-1}(x)=-\\frac{1}{3}x+\\frac{5}{3}[\/latex]\r\n\r\n4.\u00a0[latex]s^{-1}(t)=5t+10[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<div class=\"textbox learning-objectives\">\n<h1>Learning Objectives<\/h1>\n<ul>\n<li>Explain that the inverse of a linear function is a function<\/li>\n<li>Graph the inverse of a linear function<\/li>\n<li>Find the inverse function of a linear function algebraically<\/li>\n<\/ul>\n<\/div>\n<h2>The Inverse of a Function<\/h2>\n<p>We learned in section 1.3 that the inverse of a function may be obtained by exchanging the domain and range (or the input and output) of the function. For example, the inverse of the function [latex]\\{(1, 2), (3, 4), (5, 6)\\}[\/latex] is [latex]\\{(2, 1), (4, 3), (6, 5)\\}[\/latex]. All we have to do is switch the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-values in the ordered pairs. Note that the inverse of a function may or may not be a function. Figure 1 shows a function and its inverse where the inverse is a function. Figure 2 shows a function and its inverse where the inverse is NOT a function. A function will have an inverse function only if the original function is one-to-one. If the inverse of a function [latex]f[\/latex] is also a function, we may use the notation [latex]f^{-1}[\/latex] to name the inverse function. It is important to note that the &#8220;\u20131&#8221; is not an exponent. It is a superscript meaning &#8220;inverse&#8221;. The notation [latex]f^{-1}[\/latex] means the inverse function of the function [latex]f[\/latex]. This function [latex]f^{-1}[\/latex] is a completely different function from [latex]f[\/latex].<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%;\">\n<div class=\"mceTemp\">True 1 &#8211; 1 function and its inverse<\/div>\n<\/th>\n<th style=\"width: 50%;\">\n<div class=\"mceTemp\">Not a 1 &#8211; 1 function<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<div id=\"attachment_807\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-807\" class=\"wp-image-807 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/18221231\/tan-inverse-300x300.png\" alt=\"inverse functions\" width=\"300\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-807\" class=\"wp-caption-text\">Figure 1. The inverse of a 1-1 function is a 1-1 function.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">\n<div id=\"attachment_1390\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1390\" class=\"wp-image-1390 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07010603\/inverse-of-fxx%5E2-300x300.png\" alt=\"Graph of y=x^2 and its inverse\" width=\"300\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-1390\" class=\"wp-caption-text\">Figure 2. A function that is not 1-1 has an inverse that is not a function.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>The Inverse of a Linear Function<\/h2>\n<p>A linear function [latex]f(x)=mx+b[\/latex] with [latex]m\\ne0[\/latex] is a one-to-one mapping of [latex]x[\/latex] to the function value [latex]f(x)[\/latex] because each point on the graph of the function has a unique input and a unique output. There are no two points that have the same input ([latex]x[\/latex]-coordinate) or the same output ([latex]y[\/latex]-coordinate). Figure 3 illustrates some points (or solution pairs) on the graph of the function [latex]f(x)=2x[\/latex] as well as the corresponding points on the graph of the inverse function [latex]f^{-1}(x)[\/latex], which are found by simply switching the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-coordinates of the original points. Both sets of points show a one-to-one mapping. This is an example of the fact that the inverse of a linear function with a non-zero slope is also a one-to-one linear function.<\/p>\n<div id=\"attachment_801\" style=\"width: 308px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-801\" class=\"wp-image-801 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-points-298x300.png\" alt=\"One function of black points (1,2), (2,4), (3,6), (4,8), and (5,10), and the inverse function (2,1), (4,2), (6,3), (8,4), and (10,5).\" width=\"298\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-points-298x300.png 298w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-points-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-points-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-points-225x227.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-points-350x353.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/inverse-points.png 540w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><\/p>\n<p id=\"caption-attachment-801\" class=\"wp-caption-text\">Figure 3. Points (solution pairs) on the graph of the function [latex]f(x) = 2x[\/latex] and its inverse.<\/p>\n<\/div>\n<p>A linear function with a zero slope (a horizontal line), i.e., [latex]f(x)=b[\/latex], has an inverse that is not a function, [latex]x=b[\/latex] (a vertical line). A horizontal line does not represent a one-to-one function, which is why its inverse is not a function (figure 4).<\/p>\n<div id=\"attachment_1942\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1942\" class=\"wp-image-1942 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/02202149\/desmos-graph-2022-05-02T142129.550-300x300.png\" alt=\"Horizontal and vertical lines\" width=\"300\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-1942\" class=\"wp-caption-text\">Figure 4. Horizontal line is a function; vertical line is not a function.<\/p>\n<\/div>\n<h2>Graphing the Inverse of a Linear Function<\/h2>\n<p>To graph the inverse of a linear function, we may start by finding two or more points (solution pairs) on the graph of the linear function. Then we simply switch the [latex]x[\/latex]&#8211; and\u00a0[latex]y[\/latex]-coordinates of each point to find points that lie on the graph of the inverse function. We can then use these points to graph the inverse function. For example, (0, 0) and (1, 2) are two points that lie on the graph of the function [latex]f(x)=2x[\/latex]. To graph the inverse function, we switch the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-coordinates of these points to get (0, 0) and (2, 1). Therefore, the graph of the inverse function will be the line that passes through the two points (0, 0) and (2, 1) (See Figure 5).<\/p>\n<div id=\"attachment_1068\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1068\" class=\"wp-image-1068 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2-5-1-InverseGraph-2-300x300.png\" alt=\"A graph with a line through (0,0) and (2,1), with it's inverse, a line through (0,0) and (1,2).\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2-5-1-InverseGraph-2-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2-5-1-InverseGraph-2-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2-5-1-InverseGraph-2-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2-5-1-InverseGraph-2-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2-5-1-InverseGraph-2-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2-5-1-InverseGraph-2-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/2-5-1-InverseGraph-2.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1068\" class=\"wp-caption-text\">Figure 5. Graph of the function [latex]f(x)=2x[\/latex] and its inverse.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Use the graphs of the linear functions to graph their inverse function.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%;\">Problem 1<\/th>\n<th style=\"width: 50%;\">Problem 2<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">&nbsp;<\/p>\n<div id=\"attachment_1392\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1392\" class=\"wp-image-1392\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07014041\/fx4x-2-300x298.png\" alt=\"Graph of f(x)=4x-2\" width=\"300\" height=\"298\" \/><\/p>\n<p id=\"caption-attachment-1392\" class=\"wp-caption-text\">[latex]f(x)=4x-2[\/latex]<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">&nbsp;<\/p>\n<div id=\"attachment_1395\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1395\" class=\"wp-image-1395\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07015330\/fx-23x5-300x300.png\" alt=\"Graph of f(x)=-2\/3 x+5\" width=\"300\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-1395\" class=\"wp-caption-text\">[latex]f(x)=-\\frac{2}{3}x+5[\/latex]<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Solution<\/h4>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%;\">Problem 1<\/th>\n<th style=\"width: 50%;\">\n<p style=\"text-align: initial;\">Problem 2<\/p>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">&nbsp;<\/p>\n<p style=\"text-align: initial;\">Choose 2 (or more) points on the original line: (0, \u20132) and (1, 2)<\/p>\n<p style=\"text-align: initial;\">Reverse the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-coordinates: (\u20132, 0) and (2, 1)<\/p>\n<p style=\"text-align: initial;\">Plot the new points. The line that passes through them is the inverse function.<\/p>\n<div id=\"attachment_1391\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1391\" class=\"wp-image-1391\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07013646\/fx4x-2-and-inverse-300x294.png\" alt=\"Graph of f(x)=4x-2 and its inverse\" width=\"300\" height=\"294\" \/><\/p>\n<p id=\"caption-attachment-1391\" class=\"wp-caption-text\">[latex]f(x)=4x-2[\/latex] and its inverse<\/p>\n<\/div>\n<p style=\"text-align: initial;\">\n<\/td>\n<td style=\"width: 50%;\">\n<p style=\"text-align: initial;\">Choose 2 (or more) points on the original line: (0, 5) and (6, 1)<\/p>\n<p style=\"text-align: initial;\">Reverse the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-coordinates: (5, 0) and (1, 6)<\/p>\n<p style=\"text-align: initial;\">Plot the new points. The line that passes through them is the inverse function.<\/p>\n<div id=\"attachment_1396\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1396\" class=\"wp-image-1396\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07015336\/fx-23x5-and-inverse-300x296.png\" alt=\"Graph of f(x)=-2\/3 x + 5 and its inverse\" width=\"300\" height=\"296\" \/><\/p>\n<p id=\"caption-attachment-1396\" class=\"wp-caption-text\">[latex]f(x)=-\\frac{2}{3}x+5[\/latex] and its inverse<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Use the graphs of the linear functions to graph their inverse function.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">Problem 1<\/td>\n<td style=\"width: 50%;\">Problem 2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">&nbsp;<\/p>\n<div id=\"attachment_1393\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1393\" class=\"wp-image-1393\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07014325\/fx-3x5-300x297.png\" alt=\"Graph of f(x)=-3x+5\" width=\"300\" height=\"297\" \/><\/p>\n<p id=\"caption-attachment-1393\" class=\"wp-caption-text\">[latex]f(x)=-3x+5[\/latex]<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">&nbsp;<\/p>\n<div id=\"attachment_1397\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1397\" class=\"wp-image-1397\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07020047\/fx43x2-300x294.png\" alt=\"Graph of f(x)=4\/3 x+2\" width=\"300\" height=\"294\" \/><\/p>\n<p id=\"caption-attachment-1397\" class=\"wp-caption-text\">[latex]f(x)=\\frac{4}{3}x+2[\/latex]<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm853\">Show Answer<\/span><\/p>\n<div id=\"qhjm853\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">Problem 1<\/td>\n<td style=\"width: 50%;\">Problem 2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">&nbsp;<\/p>\n<div id=\"attachment_1394\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1394\" class=\"wp-image-1394\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07014620\/fx-3x5-and-inverse-300x300.png\" alt=\"Graph of f(x)=-3x+5 and its inverse\" width=\"300\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-1394\" class=\"wp-caption-text\">[latex]f(x)=-3x+5[\/latex] and its inverse<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%;\">&nbsp;<\/p>\n<div id=\"attachment_1398\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1398\" class=\"wp-image-1398\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07020052\/fx43x2-and-inverse-300x297.png\" alt=\"Graph of f(x)=4\/3 x+2 and its inverse\" width=\"300\" height=\"297\" \/><\/p>\n<p id=\"caption-attachment-1398\" class=\"wp-caption-text\">[latex]f(x)=\\frac{4}{3}x+2[\/latex] and its inverse<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Graph the function [latex]f(x)=\\frac{2}{5}x+2[\/latex], then graph its inverse.<\/p>\n<h4>Solution<\/h4>\n<p>To graph\u00a0[latex]f(x)=\\frac{2}{5}x+2[\/latex] we can either use a table of values or use the slope and [latex]y[\/latex]-intercept.<\/p>\n<p>The slope of the line is [latex]m=\\frac{2}{5}[\/latex] and the [latex]y[\/latex]-intercept is (0, 2).<\/p>\n<p>To graph the line, we start at (0, 2) then run 5 units to the right and 2 units up to get to (5, 4).<\/p>\n<div id=\"attachment_1400\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1400\" class=\"wp-image-1400\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07021055\/fx25-x2-300x291.png\" alt=\"Graph of f(x)=2\/5 x+2 showing points at (0,2) and (5,4).\" width=\"300\" height=\"291\" \/><\/p>\n<p id=\"caption-attachment-1400\" class=\"wp-caption-text\">[latex]f(x)=\\frac{2}{5}x+2[\/latex]<\/p>\n<\/div>\n<p>Now reverse the points (0, 2) and (5, 4) to (2, 0) and (4, 5) and draw the line of the inverse function through these new points.<\/p>\n<div id=\"attachment_1399\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1399\" class=\"wp-image-1399\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07021050\/fx25-x2-and-inverse-300x297.png\" alt=\"Graph of f(x)=2\/5 x+2 showing points (0,2) and (5,4), and its inverse showing points (2,0) and (4,5).\" width=\"300\" height=\"297\" \/><\/p>\n<p id=\"caption-attachment-1399\" class=\"wp-caption-text\">[latex]f(x)=\\frac{2}{5}x+2[\/latex] and its inverse<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Graph the function [latex]f(x)=\\frac{1}{3}x-4[\/latex], then graph its inverse.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm277\">Show Answer<\/span><\/p>\n<div id=\"qhjm277\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"attachment_1401\" style=\"width: 372px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1401\" class=\"wp-image-1401\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07021926\/fx13-x-4-and-inverse-300x292.png\" alt=\"Graph of f(x)=1\/3 x-4 and inverse. Slopes described below.\" width=\"362\" height=\"352\" \/><\/p>\n<p id=\"caption-attachment-1401\" class=\"wp-caption-text\">[latex]f(x)=\\frac{1}{3}x-4[\/latex] and its inverse<\/p>\n<\/div>\n<p>Notice the relationship of the slope of the original function to the slope of the inverse function. The original function has a slope of 3, while the inverse function has a slope of [latex]\\frac{1}{3}[\/latex]. i.e. the slopes are reciprocals of each other, but NOT negative reciprocals like perpendicular lines.<\/p><\/div>\n<\/div>\n<\/div>\n<h2>Inverse Functions and Symmetry<\/h2>\n<p>As we saw in chapter 1, the graph of any function and its inverse are symmetric across the line [latex]y=x[\/latex].\u00a0 For example, figure 6 shows the graph of [latex]f(x)=\\frac{2}{5}x+2[\/latex] and its inverse. With the added line [latex]y=x[\/latex], we can see that the two lines are symmetric (mirror images of one another) across the line [latex]y=x[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1409 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/07033342\/Inverse-functions-and-symmetry-300x293.png\" alt=\"Graph showing symmetry of inverse functions across the line y=x\" width=\"318\" height=\"310\" \/><\/p>\n<p style=\"text-align: center;\">Figure 6. [latex]f(x)=\\frac{2}{5}x+2[\/latex] and its inverse<\/p>\n<p>Notice the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-intercepts. The [latex]y[\/latex]-intercept of (0, 2) in the original function (blue line) reflects to the [latex]x[\/latex]-intercept (2, 0) in the inverse function (green line). Also, the\u00a0[latex]x[\/latex]-intercept of (\u20135, 0) in the original function (blue line) reflects to the [latex]y[\/latex]-intercept (0, \u20135) in the inverse function (green line).<\/p>\n<p>The slopes of each function are also related. The function [latex]f(x)[\/latex] has a slope of [latex]\\frac{2}{5}[\/latex], while the inverse function [latex]f^{-1}(x)[\/latex] has a slope of [latex]\\frac{5}{2}[\/latex]. <strong>The slopes of inverse functions are reciprocals of each other. <\/strong>This is\u00a0because the slope of a function is [latex]\\dfrac{\\text{change in y}}{\\text{change in x}}[\/latex]. The slope of the inverse function becomes\u00a0[latex]\\dfrac{\\text{change in x}}{\\text{change in y}}[\/latex].<\/p>\n<p>Since we know the slopes are reciprocals and the [latex]y[\/latex]-intercept of the inverse function is the flipped coordinates of\u00a0 the [latex]x[\/latex]-intercept of the original function, we can find the equation of the inverse function: [latex]m=\\frac{5}{2},\\;b=-5[\/latex] so [latex]f^{-1}(x)=\\frac{5}{2}x-5[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Graph the function [latex]g(x)=\\frac{3}{7}x-2[\/latex] then use symmetry to graph its inverse. Write the equation of the inverse function.<\/p>\n<h4>Solution<\/h4>\n<p>To graph\u00a0[latex]g(x)=\\frac{3}{7}x-2[\/latex], we can start at the [latex]y[\/latex]-intercept (0, \u20132) then use the slope of [latex]\\frac{3}{7}[\/latex] to run 7 and rise 3 to get to another point on the line: (7, 1). Then draw the line through these points.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1436 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/gx37-x-2-plus-slope-300x292.png\" alt=\"Graph of g(x) showing an intercept of (0,-2) with the slope of rise 3 and run 7 yielding a second point of (7,1).\" width=\"300\" height=\"292\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/gx37-x-2-plus-slope-300x292.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/gx37-x-2-plus-slope-768x748.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/gx37-x-2-plus-slope-1024x997.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/gx37-x-2-plus-slope-65x63.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/gx37-x-2-plus-slope-225x219.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/gx37-x-2-plus-slope-350x341.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/gx37-x-2-plus-slope.png 1672w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>To graph the inverse function, we can draw the line [latex]y=x[\/latex] then reflect points on the line of the original function across [latex]y=x[\/latex] to get points on the inverse function. Then we can draw the line of the inverse function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1437 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/gx37x-2-and-inverse-300x294.png\" alt=\"G(x) and its inverse. The inverse goes through (-2,0) and (1,7).\" width=\"300\" height=\"294\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/gx37x-2-and-inverse-300x294.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/gx37x-2-and-inverse-768x753.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/gx37x-2-and-inverse-1024x1004.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/gx37x-2-and-inverse-65x64.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/gx37x-2-and-inverse-225x221.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/gx37x-2-and-inverse-350x343.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/03\/gx37x-2-and-inverse.png 1660w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>To write the equation of the inverse function, we know that the slope is [latex]\\frac{7}{3}[\/latex]; the reciprocal of the slope of the original function.<\/p>\n<p>From the graph, the [latex]y[\/latex]-intercept is fractional but we know that the inverse goes through the point (\u20132, 0); the flipped point of (0 \u20132) on the original function.<\/p>\n<p>The inverse function is linear, so has the form [latex]f(x)=mx+b=\\frac{7}{3}x+b[\/latex] with the substituted slope.<\/p>\n<p>To find [latex]b[\/latex], we can use the point (\u20132, 0):<\/p>\n<p>[latex]\\begin{aligned}f(x)&=\\frac{7}{3}x+b\\\\ 0&=\\frac{7}{3}\\cdot (-2)+b\\\\ 0&=\\frac{-14}{3}+b\\\\b&= \\frac{14}{3}\\end{aligned}[\/latex]<\/p>\n<p>Consequently the inverse function is [latex]g^{-1}(x)=\\frac{7}{3}x+\\frac{14}{3}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Graph the function [latex]g(x)=\\frac{1}{2}x+3[\/latex] then use symmetry to graph its inverse. Write the equation of the inverse function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm840\">Show Answer<\/span><\/p>\n<div id=\"qhjm840\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1438\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/03\/08202803\/gx12-x3-and-inverse-300x296.png\" alt=\"Graph of g(x) with y intercept (0,3) and slope of 1\/2,and its inverse with x intercept of (3,0) and slope of 2.\" width=\"405\" height=\"400\" \/><\/p>\n<p style=\"text-align: center;\">[latex]g^{-1}(x)=2x-6[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Figure 7 demonstrates the symmetry across the line [latex]y=x[\/latex] of a function and its inverse. Each point on the line of the original function has a reflected point across [latex]y=x[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" src=\"https:\/\/www.desmos.com\/calculator\/opmximd6d0?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><\/iframe><\/p>\n<p style=\"text-align: center;\">Figure 7. The symmetry of a function and its inverse.<\/p>\n<h2>The Algebraic Inverse Function of a Linear Function<\/h2>\n<p>We have learned that to graph an inverse function, we simply switch the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-coordinates on the graph. The same is true for finding the inverse function algebraically. To find the inverse function of a given linear function, we switch the independent variable (e.g.,\u00a0[latex]x[\/latex]) to [latex]y[\/latex] so that it becomes the dependent variable, and the dependent variable (e.g.,\u00a0[latex]y=f(x)[\/latex]) to [latex]x[\/latex] to become the independent variable. The inverse of a function is the exchange of the domain and range of the function.<\/p>\n<p>From the perspective of an equation, a linear equation in the form [latex]y=mx+b[\/latex], becomes its inverse by switching [latex]x[\/latex] and [latex]y[\/latex] to get [latex]x=my+b[\/latex].<\/p>\n<p>For example, if\u00a0[latex]y = 2x + 1[\/latex], the inverse is [latex]x=2y+1[\/latex].<\/p>\n<p>For the function\u00a0[latex]f(x) = 2x + 1[\/latex], we can start by writing [latex]y=f(x)[\/latex] so that\u00a0[latex]y = 2x + 1[\/latex]. Then it is easier to switch the variables to find the inverse\u00a0[latex]x = 2y + 1[\/latex].<\/p>\n<p>To write this inverse equation as a function, we need to solve it for [latex]y[\/latex] by first subtracting 1 from both sides, then dividing both sides by 2:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{equation}\\begin{aligned}x &= 2y+1\\\\x-1&=2y\\\\\\frac{x-1}{2}&=y\\\\\\frac{1}{2}x-\\frac{1}{2}&=y\\end{aligned}\\end{equation}[\/latex]<\/p>\n<p>The last step for finding the inverse function is to use function notation and write [latex]f^{-1}(x)=y[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]f^{-1}(x) = \\frac{1}{2}x - \\frac{1}{2}[\/latex]<\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">This method of switching [latex]x[\/latex] and [latex]y[\/latex] is based on what we learned through graphing functions and their inverses on the coordinate plane, where the independent variable is always represented by the variable [latex]x[\/latex] and the dependent variable is always represented by the variable [latex]y[\/latex]. In other words, the\u00a0[latex]x-axis[\/latex] represents the independent variable and the\u00a0[latex]y-axis[\/latex] the dependent variable. The final step of replacing\u00a0[latex]y[\/latex] with\u00a0[latex]f^{-1}(x)[\/latex] is because the inverse is a function related to the original function and although we need to name it differently from the original function [latex]f(x)[\/latex], we want to keep the relationship this inverse function has to the original function [latex]f(x)[\/latex].<\/span><\/p>\n<p>Remember that in function notation the function name does not have to be [latex]f[\/latex], nor does the independent variable have to be [latex]x[\/latex]. It&#8217;s only when graphing that [latex]x[\/latex] and [latex]y[\/latex] are typically used.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>Find the inverse of the linear function:<\/p>\n<ol>\n<li>[latex]f(x)=2x[\/latex]<\/li>\n<li>[latex]g(x)=x+6[\/latex]<\/li>\n<li>[latex]h(x)=4x+5[\/latex]<\/li>\n<li>[latex]s(t)=\\frac{3}{4}t-2[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<p>Start by replacing the function name with [latex]y[\/latex]; switch the variables; solve for [latex]y[\/latex]; write inverse function notation for [latex]y[\/latex].<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\">\n<p style=\"text-align: initial;\">Finding an inverse<\/p>\n<\/td>\n<td style=\"width: 50%;\">Four basic steps<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">\n<p style=\"text-align: initial;\">1.<\/p>\n<p style=\"text-align: initial;\">[latex]\\begin{equation}\\begin{aligned}f(x) &=2x\\\\y &=2x\\\\x &=2y\\\\ \\frac{1}{2}x &=y \\\\f^{-1}(x) &=\\frac{1}{2}x\\end{aligned}\\end{equation}[\/latex]<\/p>\n<\/td>\n<td style=\"width: 50%;\">2.<\/p>\n<p>[latex]\\begin{equation}\\begin{aligned}g(x)&=x+6\\\\y &=x+6\\\\x &=y+6\\\\ x-6 &=y \\\\g^{-1}(x) &=x-6\\end{aligned}\\end{equation}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">3.<\/p>\n<p>[latex]\\begin{equation}\\begin{aligned}h(x)&=4x+5\\\\y &=4x+5\\\\x &=4y+5\\\\x-5 &=4y\\\\ \\frac{x-5}{4}&=y \\\\h^{-1}(x) &=\\frac{1}{4}x-\\frac{5}{4}\\end{aligned}\\end{equation}[\/latex]<\/td>\n<td style=\"width: 50%;\">4.<\/p>\n<p>[latex]\\begin{equation}\\begin{aligned}s(t)&=\\frac{3}{4}t-2\\\\y &=\\frac{3}{4}t-2\\\\t &=\\frac{3}{4}y-2\\\\ t+2 &=\\frac{3}{4}y\\\\ \\frac{4}{3}(t+2)&=y \\\\s^{-1}(t) &=\\frac{4}{3}t+\\frac{8}{3}\\end{aligned}\\end{equation}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>Find the inverse of the linear function:<\/p>\n<ol>\n<li>[latex]f(x)=-6x[\/latex]<\/li>\n<li>[latex]g(x)=2x-1[\/latex]<\/li>\n<li>[latex]h(x)=-3x+5[\/latex]<\/li>\n<li>[latex]s(t)=\\frac{1}{5}t-2[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm143\">Show Answer<\/span><\/p>\n<div id=\"qhjm143\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. [latex]f^{-1}(x)=-\\frac{1}{6}x[\/latex]<\/p>\n<p>2.\u00a0[latex]g^{-1}(x)=\\frac{1}{2}x+\\frac{1}{2}[\/latex]<\/p>\n<p>3.\u00a0[latex]h^{-1}(x)=-\\frac{1}{3}x+\\frac{5}{3}[\/latex]<\/p>\n<p>4.\u00a0[latex]s^{-1}(t)=5t+10[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1055\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>The Inverse of a Linear Function . <strong>Authored by<\/strong>: Hazel McKenna and Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All graphs created using desmos graphing calculator. <strong>Authored by<\/strong>: Hazel McKenna and Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.desmos.com\/calculator\">http:\/\/www.desmos.com\/calculator<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All Examples and Try Its. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Figure 7. . <strong>Authored by<\/strong>: John Jarvis. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":422608,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"The Inverse of a Linear Function \",\"author\":\"Hazel McKenna and Leo Chang\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All graphs created using desmos graphing calculator\",\"author\":\"Hazel McKenna and Leo Chang\",\"organization\":\"Utah Valley 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