{"id":2264,"date":"2022-05-20T21:29:29","date_gmt":"2022-05-20T21:29:29","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=2264"},"modified":"2026-01-22T20:02:07","modified_gmt":"2026-01-22T20:02:07","slug":"5-5-the-inverse-function-of-an-exponential-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/5-5-the-inverse-function-of-an-exponential-function\/","title":{"raw":"5.5 The Inverse of an Exponential Function","rendered":"5.5 The Inverse of an Exponential 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>Graph the inverse function of an exponential function<\/li>\r\n \t<li>Determine the equation of the inverse function of an exponential function<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn chapter 3, we discussed that every function has an inverse, but <strong>only a one-to-one function has an inverse function<\/strong>. Since an exponential function is a one-to-one function, its inverse is also a one-to-one function. Therefore, the inverse of an exponential function is also a function.\r\n<h2>Graphing the Inverse Function of an Exponential Function<\/h2>\r\nWe can graph the inverse of an exponential function by creating and using a table of values. For example, given the function [latex]f(x)=2^x[\/latex], we may graph the function by creating a table of values by choosing [latex]x[\/latex]-values then determining the corresponding function values (table 1).\r\n<table class=\"aligncenter\" style=\"border-collapse: collapse; width: 50%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 10.9756%; text-align: center;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 6.09756%; text-align: center;\">[latex]y=f(x)=2^x[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 10.9756%; text-align: center;\">[latex]-3[\/latex]<\/td>\r\n<td style=\"width: 6.09756%; text-align: center;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 10.9756%; text-align: center;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 6.09756%; text-align: center;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 10.9756%; text-align: center;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 6.09756%; text-align: center;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 10.9756%; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 6.09756%; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 10.9756%; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 6.09756%; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 10.9756%; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 6.09756%; text-align: center;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 10.9756%; text-align: center;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 6.09756%; text-align: center;\">[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 17.0732%;\" colspan=\"2\">Table 1. Table of values<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: left;\">The inverse of the function is found by switching the values of the [latex]x[\/latex] and [latex]y[\/latex] columns so that the inputs are the values of [latex]y[\/latex] and the outputs are the values of [latex]x[\/latex]. Table 2 shows the values after switching the [latex]x[\/latex] and [latex]y[\/latex] columns, and is called the inverse table.<\/p>\r\n\r\n<table class=\"aligncenter\" style=\"border-collapse: collapse; width: 50%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 2.42718%; text-align: center;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 2.42718%; text-align: center;\">[latex]y=f^{-1}(x)[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 2.42718%; text-align: center;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\r\n<td style=\"width: 2.42718%; text-align: center;\">[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 2.42718%; text-align: center;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<td style=\"width: 2.42718%; text-align: center;\">[latex]-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 2.42718%; text-align: center;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td style=\"width: 2.42718%; text-align: center;\">[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 2.42718%; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 2.42718%; text-align: center;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 2.42718%; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 2.42718%; text-align: center;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 2.42718%; text-align: center;\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 2.42718%; text-align: center;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 2.42718%; text-align: center;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 2.42718%; text-align: center;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 4.85436%;\" colspan=\"2\">Table 2. Inverse table<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFigure 1 shows the graphs of the function [latex]f(x)=2^x[\/latex] (blue curve) and its inverse function (green curve) based on the values in table 1 of the function and its inverse table (table 2). Notice that the graph of the inverse function is a reflection of the graph of the original function with respect to the line of symmetry [latex]y=x[\/latex] (red line).\r\n\r\n[caption id=\"attachment_3246\" align=\"aligncenter\" width=\"353\"]<img class=\"wp-image-3246\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/08170857\/5-5-InverseNew-300x300.png\" alt=\"f(x)=2^x and its inverse graphed\" width=\"353\" height=\"353\" \/> Figure 1. The graphs of the function f(x)=2^x and its inverse function [latex]x=2^y[\/latex][\/caption]Notice that any point [latex](x, y)[\/latex] on the original function becomes the point [latex](y, x)[\/latex] on the inverse function. The domain of the original function becomes the range of the inverse function, and vice versa. Domain of [latex]f(x)=(-\\infty, +\\infty)=[\/latex] range of [latex]f^{-1}(x)[\/latex]. Range of\u00a0[latex]f(x)=(0, +\\infty)=[\/latex] domain of [latex]f^{-1}(x)[\/latex].\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nUse Desmos to graph the function [latex]f(x)=2^{x-3}[\/latex] then use reflection about the line [latex]y=x[\/latex] to graph the inverse function [latex]y=f^{-1}(x)[\/latex].\r\n<h4>Solution<\/h4>\r\nPutting the function\u00a0[latex]f(x)=2^{x-3}[\/latex] into Desmos gives the graph:\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-2923 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-21T190830.360-300x300.png\" alt=\"y=2^(x-3)\" width=\"300\" height=\"300\" \/><\/p>\r\nDraw the line [latex]y=x[\/latex] then reflect the curve across the line so that any point [latex](x, y)[\/latex] reflects to [latex](y, x)[\/latex].\r\n<p style=\"text-align: center;\"><img class=\"aligncenter wp-image-2922 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/22010704\/desmos-graph-2022-06-21T190621.772-300x300.png\" alt=\"y=2^(x-3) and inverse\" width=\"300\" height=\"300\" \/><\/p>\r\n<p style=\"text-align: center;\"><\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nUse Desmos to graph the function [latex]f(x)=3^{x+4}+2[\/latex] then use reflection about the line [latex]y=x[\/latex] to graph the inverse function [latex]y=f^{-1}(x)[\/latex].\r\n\r\n[reveal-answer q=\"hjm286\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm286\"]\r\n<p style=\"text-align: center;\"><img class=\"aligncenter size-medium wp-image-2929\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/22023736\/desmos-graph-2022-06-21T203658.162-300x300.png\" alt=\"exponential function and its inverse\" width=\"300\" height=\"300\" \/><\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nAn exponential function has domain [latex](-\\infty, +\\infty)[\/latex] and range [latex](2, +\\infty)[\/latex]. What is the domain and range of its inverse function?\r\n<h4>Solution<\/h4>\r\nThe range of the original function becomes the domain of the inverse function and vice versa.\r\n\r\nSo, the domain of the inverse function is\u00a0[latex](2, +\\infty)[\/latex] and the range of the inverse function is\u00a0[latex](-\\infty, +\\infty)[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nAn exponential function has domain [latex](-\\infty, +\\infty)[\/latex] and range [latex](-3, +\\infty)[\/latex]. What is the domain and range of its inverse function?\r\n\r\n[reveal-answer q=\"hjm765\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm765\"]\r\n\r\nDomain of the inverse function is\u00a0[latex](-3, +\\infty)[\/latex] and the range of the inverse function is\u00a0[latex](-\\infty, +\\infty)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Determining the Equation of the Inverse Function<\/h2>\r\nThe inverse function of an exponential function [latex]f(x)=r^x[\/latex], is found by switching the input [latex]x[\/latex] and output [latex]y[\/latex]. We start by writing [latex]y[\/latex] for [latex]f(x)[\/latex] then switch [latex]x[\/latex] and [latex]y[\/latex] to get the inverse function:\r\n<p style=\"text-align: center;\">Original function: [latex]y=r^x[\/latex]<\/p>\r\n<p style=\"text-align: center;\">Inverse function:\u00a0[latex]x=r^y[\/latex]<\/p>\r\nThe minute we switch\u00a0[latex]x[\/latex] and [latex]y[\/latex], we have the inverse function. Now all we have to do is solve for [latex]y[\/latex] so we can write the inverse function using function notation.\r\n\r\nTo solve the equation\u00a0[latex]x=r^y[\/latex], we need to solve for the exponent. To do this we introduce logarithms.\u00a0Logarithms were invented by John Napier, a Scottish mathematician, in\u00a01614 as a means of simplifying calculations.\r\n<div class=\"textbox shaded\">\r\n<h3>logarithms<\/h3>\r\nA logarithm is a quantity representing the power to which a fixed number (the base) must be raised to produce a given number.\r\n<p style=\"text-align: center;\">For all real numbers [latex]x[\/latex] and positive real numbers [latex]a[\/latex] and<span style=\"font-size: 1rem;\">\u00a0[latex]b, b\\neq1[\/latex],<\/span><\/p>\r\n<p style=\"text-align: center;\">if [latex]a=b^x[\/latex], then [latex]x=log_b{a}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]x=log_b{a}[\/latex] is read '[latex]x[\/latex] equals the logarithm base [latex]b[\/latex] of [latex]a[\/latex].<\/p>\r\n<p style=\"text-align: center;\">The base of the exponential expression becomes the base of the logarithm.<\/p>\r\n\r\n<\/div>\r\n<div>Some logarithms are easy to find. For example, [latex]log_{10}{100}=2[\/latex] because [latex]10^2=100[\/latex], and [latex]log_3{81}=4[\/latex] because [latex]3^4=81[\/latex]. Other logarithms are much more complicated. For example, to find the value of [latex]log_4{9}[\/latex] we need to find a number [latex]x[\/latex] such that [latex]4^x=9[\/latex]. Since [latex]4^1=4[\/latex] and [latex]4^2=16[\/latex] and 4 &lt; 9 &lt; 16, [latex]x[\/latex] is somewhere between 1 and 2.\u00a0 In fact, it is approximately 1.486. i.e.[latex]4^{1.486}=9[\/latex]. Luckily, as we will see in chapter 6, our calculators will do the heavy lifting for us, as John Napier and others figured out all the logarithms a long time ago.<\/div>\r\n&nbsp;\r\n<div>The inverse function of\u00a0\u00a0[latex]y=r^x[\/latex] is\u00a0[latex]x=r^y[\/latex]. Now let's solve for [latex]y[\/latex].<\/div>\r\n<div>By the definition of logarithm,<\/div>\r\n<div style=\"text-align: center;\">[latex]x=r^y[\/latex]<\/div>\r\n<div>is equivalent to,<\/div>\r\n<div style=\"text-align: center;\">[latex]y=log_r{x}[\/latex]<\/div>\r\n<div>Then we can write the inverse in function notation,<\/div>\r\n<div style=\"text-align: center;\">[latex]f^{-1}(x)=log_r{x}[\/latex]<\/div>\r\n&nbsp;\r\n<div>We will learn more about logarithms in chapter 6. For now, we just need to know that a logarithm is by definition the exponent to which a base must be raised to produce a given number. In addition, the logarithm function is the inverse of the exponential function.<\/div>\r\n<div>\r\n<div class=\"textbox shaded\">\r\n<h3>INVERSE FUNCTION OF AN EXPONENTIAL FUNCTION<\/h3>\r\n<p style=\"text-align: center;\">The exponential function [latex]f(x)=r^x[\/latex] has an inverse function [latex]f^{-1}(x)=log_r{x}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nDetermine the inverse function of [latex]f(x)=5^{x-1}[\/latex].\r\n<h4>Solution<\/h4>\r\nThe inverse function is found by switching [latex]x[\/latex] and [latex]y[\/latex] in the function [latex]y=5^{x-1}[\/latex]:\r\n<p style=\"text-align: center;\">[latex]x=5^{y-1}[\/latex]<\/p>\r\nNow we need to solve for [latex]y[\/latex]. By definition, if\u00a0[latex]a=b^x[\/latex], then [latex]x=log_b{a}[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{aligned}x&amp;=5^{y-1}\\\\y-1&amp;=log_5{x}\\\\y&amp;=log_5{x}+1\\end{aligned}[\/latex]<\/p>\r\nFinish by writing the inverse in function notation. Replace [latex]y[\/latex] with [latex]f^{-1}(x)[\/latex]:\r\n<p style=\"text-align: center;\">[latex]f^{-1}(x)=log_5(x)+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nDetermine the inverse function of\u00a0[latex]g(x)=5^x-1[\/latex].\r\n<h4>Solution<\/h4>\r\nThe inverse function is found by switching [latex]x[\/latex]and [latex]y[\/latex] in the function [latex]y=5^x-1[\/latex]:\r\n<p style=\"text-align: center;\">[latex]x=5^y-1[\/latex]<\/p>\r\nNow we need to solve for [latex]y[\/latex]. By definition, if\u00a0[latex]a=b^x[\/latex], then [latex]x=log_b{a}[\/latex].\r\n\r\nWe start by isolating the exponential term in the equation by adding 1 to both sides:\r\n<p style=\"text-align: center;\">[latex]x+1=5^y[\/latex]<\/p>\r\nUse the definition of logarithms:\r\n<p style=\"text-align: center;\">[latex]y=log_5{(x+1)}[\/latex]<\/p>\r\nFinish by writing the equation in function notation:\r\n<p style=\"text-align: center;\">[latex]g^{-1}(x)=log_5{(x+1)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 5<\/h3>\r\nDetermine the inverse function of\u00a0[latex]h(x)=(3)2^x+5[\/latex].\r\n<h4>Solution<\/h4>\r\nSwitch [latex]x[\/latex] and [latex]y[\/latex] in [latex]y=(3)2^x+5[\/latex]:\r\n<p style=\"text-align: center;\">[latex]x=(3)2^y+5[\/latex]<\/p>\r\nIsolate the exponential term by subtracting 5 from both sides:\r\n<p style=\"text-align: center;\">[latex]x-5=(3)2^y[\/latex]<\/p>\r\nIsolate the exponential by dividing both sides by 3:\r\n<p style=\"text-align: center;\">[latex]\\dfrac{x-5}{3}=2^y[\/latex]<\/p>\r\nUse the definition of logarithms to solve for [latex]y[\/latex]:\r\n<p style=\"text-align: center;\">[latex]y=log_2{\\left(\\dfrac{x-5}{3}\\right)}[\/latex]<\/p>\r\nWrite in function notation:\r\n<p style=\"text-align: center;\">[latex]h^{-1}(x)=log_2{\\left(\\dfrac{x-5}{3}\\right)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nDetermine the inverse function of [latex]f(x)=4^{x+3}[\/latex].\r\n\r\n[reveal-answer q=\"hjm487\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm487\"]\r\n\r\n[latex]f^{-1}(x)=log_4{x}-3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nDetermine the inverse function of\u00a0[latex]g(x)=7^x-4[\/latex].\r\n\r\n[reveal-answer q=\"142680\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"142680\"]\r\n\r\n[latex]g^{-1}(x)=log_7{(x+4)}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nDetermine the inverse function of\u00a0[latex]h(x)=(5)6^x-4[\/latex].\r\n\r\n[reveal-answer q=\"hjm973\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm973\"]\r\n\r\n[latex]h^{-1}(x)=log_6{\\left(\\dfrac{x+4}{5}\\right)}[\/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>Graph the inverse function of an exponential function<\/li>\n<li>Determine the equation of the inverse function of an exponential function<\/li>\n<\/ul>\n<\/div>\n<p>In chapter 3, we discussed that every function has an inverse, but <strong>only a one-to-one function has an inverse function<\/strong>. Since an exponential function is a one-to-one function, its inverse is also a one-to-one function. Therefore, the inverse of an exponential function is also a function.<\/p>\n<h2>Graphing the Inverse Function of an Exponential Function<\/h2>\n<p>We can graph the inverse of an exponential function by creating and using a table of values. For example, given the function [latex]f(x)=2^x[\/latex], we may graph the function by creating a table of values by choosing [latex]x[\/latex]-values then determining the corresponding function values (table 1).<\/p>\n<table class=\"aligncenter\" style=\"border-collapse: collapse; width: 50%;\">\n<tbody>\n<tr>\n<th style=\"width: 10.9756%; text-align: center;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 6.09756%; text-align: center;\">[latex]y=f(x)=2^x[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 10.9756%; text-align: center;\">[latex]-3[\/latex]<\/td>\n<td style=\"width: 6.09756%; text-align: center;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 10.9756%; text-align: center;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 6.09756%; text-align: center;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 10.9756%; text-align: center;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 6.09756%; text-align: center;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 10.9756%; text-align: center;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 6.09756%; text-align: center;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 10.9756%; text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 6.09756%; text-align: center;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 10.9756%; text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 6.09756%; text-align: center;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 10.9756%; text-align: center;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 6.09756%; text-align: center;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 17.0732%;\" colspan=\"2\">Table 1. Table of values<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: left;\">The inverse of the function is found by switching the values of the [latex]x[\/latex] and [latex]y[\/latex] columns so that the inputs are the values of [latex]y[\/latex] and the outputs are the values of [latex]x[\/latex]. Table 2 shows the values after switching the [latex]x[\/latex] and [latex]y[\/latex] columns, and is called the inverse table.<\/p>\n<table class=\"aligncenter\" style=\"border-collapse: collapse; width: 50%;\">\n<tbody>\n<tr>\n<th style=\"width: 2.42718%; text-align: center;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 2.42718%; text-align: center;\">[latex]y=f^{-1}(x)[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 2.42718%; text-align: center;\">[latex]\\dfrac{1}{8}[\/latex]<\/td>\n<td style=\"width: 2.42718%; text-align: center;\">[latex]-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 2.42718%; text-align: center;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<td style=\"width: 2.42718%; text-align: center;\">[latex]-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 2.42718%; text-align: center;\">[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td style=\"width: 2.42718%; text-align: center;\">[latex]-1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 2.42718%; text-align: center;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 2.42718%; text-align: center;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 2.42718%; text-align: center;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 2.42718%; text-align: center;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 2.42718%; text-align: center;\">[latex]4[\/latex]<\/td>\n<td style=\"width: 2.42718%; text-align: center;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 2.42718%; text-align: center;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 2.42718%; text-align: center;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 4.85436%;\" colspan=\"2\">Table 2. Inverse table<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Figure 1 shows the graphs of the function [latex]f(x)=2^x[\/latex] (blue curve) and its inverse function (green curve) based on the values in table 1 of the function and its inverse table (table 2). Notice that the graph of the inverse function is a reflection of the graph of the original function with respect to the line of symmetry [latex]y=x[\/latex] (red line).<\/p>\n<div id=\"attachment_3246\" style=\"width: 363px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3246\" class=\"wp-image-3246\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/08170857\/5-5-InverseNew-300x300.png\" alt=\"f(x)=2^x and its inverse graphed\" width=\"353\" height=\"353\" \/><\/p>\n<p id=\"caption-attachment-3246\" class=\"wp-caption-text\">Figure 1. The graphs of the function f(x)=2^x and its inverse function [latex]x=2^y[\/latex]<\/p>\n<\/div>\n<p>Notice that any point [latex](x, y)[\/latex] on the original function becomes the point [latex](y, x)[\/latex] on the inverse function. The domain of the original function becomes the range of the inverse function, and vice versa. Domain of [latex]f(x)=(-\\infty, +\\infty)=[\/latex] range of [latex]f^{-1}(x)[\/latex]. Range of\u00a0[latex]f(x)=(0, +\\infty)=[\/latex] domain of [latex]f^{-1}(x)[\/latex].<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Use Desmos to graph the function [latex]f(x)=2^{x-3}[\/latex] then use reflection about the line [latex]y=x[\/latex] to graph the inverse function [latex]y=f^{-1}(x)[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>Putting the function\u00a0[latex]f(x)=2^{x-3}[\/latex] into Desmos gives the graph:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2923 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-21T190830.360-300x300.png\" alt=\"y=2^(x-3)\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-21T190830.360-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-21T190830.360-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-21T190830.360-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-21T190830.360-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-21T190830.360-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-21T190830.360-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-21T190830.360.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Draw the line [latex]y=x[\/latex] then reflect the curve across the line so that any point [latex](x, y)[\/latex] reflects to [latex](y, x)[\/latex].<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2922 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/22010704\/desmos-graph-2022-06-21T190621.772-300x300.png\" alt=\"y=2^(x-3) and inverse\" width=\"300\" height=\"300\" \/><\/p>\n<p style=\"text-align: center;\">\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Use Desmos to graph the function [latex]f(x)=3^{x+4}+2[\/latex] then use reflection about the line [latex]y=x[\/latex] to graph the inverse function [latex]y=f^{-1}(x)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm286\">Show Answer<\/span><\/p>\n<div id=\"qhjm286\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-2929\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/22023736\/desmos-graph-2022-06-21T203658.162-300x300.png\" alt=\"exponential function and its inverse\" width=\"300\" height=\"300\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>An exponential function has domain [latex](-\\infty, +\\infty)[\/latex] and range [latex](2, +\\infty)[\/latex]. What is the domain and range of its inverse function?<\/p>\n<h4>Solution<\/h4>\n<p>The range of the original function becomes the domain of the inverse function and vice versa.<\/p>\n<p>So, the domain of the inverse function is\u00a0[latex](2, +\\infty)[\/latex] and the range of the inverse function is\u00a0[latex](-\\infty, +\\infty)[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>An exponential function has domain [latex](-\\infty, +\\infty)[\/latex] and range [latex](-3, +\\infty)[\/latex]. What is the domain and range of its inverse function?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm765\">Show Answer<\/span><\/p>\n<div id=\"qhjm765\" class=\"hidden-answer\" style=\"display: none\">\n<p>Domain of the inverse function is\u00a0[latex](-3, +\\infty)[\/latex] and the range of the inverse function is\u00a0[latex](-\\infty, +\\infty)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Determining the Equation of the Inverse Function<\/h2>\n<p>The inverse function of an exponential function [latex]f(x)=r^x[\/latex], is found by switching the input [latex]x[\/latex] and output [latex]y[\/latex]. We start by writing [latex]y[\/latex] for [latex]f(x)[\/latex] then switch [latex]x[\/latex] and [latex]y[\/latex] to get the inverse function:<\/p>\n<p style=\"text-align: center;\">Original function: [latex]y=r^x[\/latex]<\/p>\n<p style=\"text-align: center;\">Inverse function:\u00a0[latex]x=r^y[\/latex]<\/p>\n<p>The minute we switch\u00a0[latex]x[\/latex] and [latex]y[\/latex], we have the inverse function. Now all we have to do is solve for [latex]y[\/latex] so we can write the inverse function using function notation.<\/p>\n<p>To solve the equation\u00a0[latex]x=r^y[\/latex], we need to solve for the exponent. To do this we introduce logarithms.\u00a0Logarithms were invented by John Napier, a Scottish mathematician, in\u00a01614 as a means of simplifying calculations.<\/p>\n<div class=\"textbox shaded\">\n<h3>logarithms<\/h3>\n<p>A logarithm is a quantity representing the power to which a fixed number (the base) must be raised to produce a given number.<\/p>\n<p style=\"text-align: center;\">For all real numbers [latex]x[\/latex] and positive real numbers [latex]a[\/latex] and<span style=\"font-size: 1rem;\">\u00a0[latex]b, b\\neq1[\/latex],<\/span><\/p>\n<p style=\"text-align: center;\">if [latex]a=b^x[\/latex], then [latex]x=log_b{a}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]x=log_b{a}[\/latex] is read &#8216;[latex]x[\/latex] equals the logarithm base [latex]b[\/latex] of [latex]a[\/latex].<\/p>\n<p style=\"text-align: center;\">The base of the exponential expression becomes the base of the logarithm.<\/p>\n<\/div>\n<div>Some logarithms are easy to find. For example, [latex]log_{10}{100}=2[\/latex] because [latex]10^2=100[\/latex], and [latex]log_3{81}=4[\/latex] because [latex]3^4=81[\/latex]. Other logarithms are much more complicated. For example, to find the value of [latex]log_4{9}[\/latex] we need to find a number [latex]x[\/latex] such that [latex]4^x=9[\/latex]. Since [latex]4^1=4[\/latex] and [latex]4^2=16[\/latex] and 4 &lt; 9 &lt; 16, [latex]x[\/latex] is somewhere between 1 and 2.\u00a0 In fact, it is approximately 1.486. i.e.[latex]4^{1.486}=9[\/latex]. Luckily, as we will see in chapter 6, our calculators will do the heavy lifting for us, as John Napier and others figured out all the logarithms a long time ago.<\/div>\n<p>&nbsp;<\/p>\n<div>The inverse function of\u00a0\u00a0[latex]y=r^x[\/latex] is\u00a0[latex]x=r^y[\/latex]. Now let&#8217;s solve for [latex]y[\/latex].<\/div>\n<div>By the definition of logarithm,<\/div>\n<div style=\"text-align: center;\">[latex]x=r^y[\/latex]<\/div>\n<div>is equivalent to,<\/div>\n<div style=\"text-align: center;\">[latex]y=log_r{x}[\/latex]<\/div>\n<div>Then we can write the inverse in function notation,<\/div>\n<div style=\"text-align: center;\">[latex]f^{-1}(x)=log_r{x}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<div>We will learn more about logarithms in chapter 6. For now, we just need to know that a logarithm is by definition the exponent to which a base must be raised to produce a given number. In addition, the logarithm function is the inverse of the exponential function.<\/div>\n<div>\n<div class=\"textbox shaded\">\n<h3>INVERSE FUNCTION OF AN EXPONENTIAL FUNCTION<\/h3>\n<p style=\"text-align: center;\">The exponential function [latex]f(x)=r^x[\/latex] has an inverse function [latex]f^{-1}(x)=log_r{x}[\/latex].<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Determine the inverse function of [latex]f(x)=5^{x-1}[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>The inverse function is found by switching [latex]x[\/latex] and [latex]y[\/latex] in the function [latex]y=5^{x-1}[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]x=5^{y-1}[\/latex]<\/p>\n<p>Now we need to solve for [latex]y[\/latex]. By definition, if\u00a0[latex]a=b^x[\/latex], then [latex]x=log_b{a}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{aligned}x&=5^{y-1}\\\\y-1&=log_5{x}\\\\y&=log_5{x}+1\\end{aligned}[\/latex]<\/p>\n<p>Finish by writing the inverse in function notation. Replace [latex]y[\/latex] with [latex]f^{-1}(x)[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]f^{-1}(x)=log_5(x)+1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>Determine the inverse function of\u00a0[latex]g(x)=5^x-1[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>The inverse function is found by switching [latex]x[\/latex]and [latex]y[\/latex] in the function [latex]y=5^x-1[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]x=5^y-1[\/latex]<\/p>\n<p>Now we need to solve for [latex]y[\/latex]. By definition, if\u00a0[latex]a=b^x[\/latex], then [latex]x=log_b{a}[\/latex].<\/p>\n<p>We start by isolating the exponential term in the equation by adding 1 to both sides:<\/p>\n<p style=\"text-align: center;\">[latex]x+1=5^y[\/latex]<\/p>\n<p>Use the definition of logarithms:<\/p>\n<p style=\"text-align: center;\">[latex]y=log_5{(x+1)}[\/latex]<\/p>\n<p>Finish by writing the equation in function notation:<\/p>\n<p style=\"text-align: center;\">[latex]g^{-1}(x)=log_5{(x+1)}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 5<\/h3>\n<p>Determine the inverse function of\u00a0[latex]h(x)=(3)2^x+5[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>Switch [latex]x[\/latex] and [latex]y[\/latex] in [latex]y=(3)2^x+5[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]x=(3)2^y+5[\/latex]<\/p>\n<p>Isolate the exponential term by subtracting 5 from both sides:<\/p>\n<p style=\"text-align: center;\">[latex]x-5=(3)2^y[\/latex]<\/p>\n<p>Isolate the exponential by dividing both sides by 3:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{x-5}{3}=2^y[\/latex]<\/p>\n<p>Use the definition of logarithms to solve for [latex]y[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]y=log_2{\\left(\\dfrac{x-5}{3}\\right)}[\/latex]<\/p>\n<p>Write in function notation:<\/p>\n<p style=\"text-align: center;\">[latex]h^{-1}(x)=log_2{\\left(\\dfrac{x-5}{3}\\right)}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Determine the inverse function of [latex]f(x)=4^{x+3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm487\">Show Answer<\/span><\/p>\n<div id=\"qhjm487\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f^{-1}(x)=log_4{x}-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>Determine the inverse function of\u00a0[latex]g(x)=7^x-4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q142680\">Show Answer<\/span><\/p>\n<div id=\"q142680\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]g^{-1}(x)=log_7{(x+4)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>Determine the inverse function of\u00a0[latex]h(x)=(5)6^x-4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm973\">Show Answer<\/span><\/p>\n<div id=\"qhjm973\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]h^{-1}(x)=log_6{\\left(\\dfrac{x+4}{5}\\right)}[\/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-2264\">\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 an Exponential Function . <strong>Authored by<\/strong>: Leo Chang and 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>All graphs created using desmos graphing calculator. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: www.desmos.com\/calculator. <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 an Exponential Function \",\"author\":\"Leo Chang and Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All graphs created using desmos graphing calculator\",\"author\":\"Hazel McKenna\",\"organization\":\"www.desmos.com\/calculator\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2264","chapter","type-chapter","status-publish","hentry"],"part":2116,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2264","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/users\/422608"}],"version-history":[{"count":42,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2264\/revisions"}],"predecessor-version":[{"id":4861,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2264\/revisions\/4861"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/2116"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2264\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=2264"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2264"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=2264"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=2264"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}