{"id":46,"date":"2025-02-13T22:43:08","date_gmt":"2025-02-13T22:43:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/inverse-functions\/"},"modified":"2026-01-08T20:23:01","modified_gmt":"2026-01-08T20:23:01","slug":"inverse-functions","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/inverse-functions\/","title":{"raw":"Inverse Functions: Learn It 1","rendered":"Inverse Functions: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Use composition to check if two functions are inverses<\/li>\r\n \t<li>Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one<\/li>\r\n \t<li>Find or calculate the inverse of a function<\/li>\r\n \t<li>Use the graph of a function to draw its inverse<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Inverse Function<\/h2>\r\nBetty is traveling to Milan for a fashion show and wants to know what the temperature will be. She is not familiar with the Celsius scale. To get an idea of how temperature measurements are related, Betty wants to convert 75 degrees Fahrenheit to degrees Celsius using the formula\r\n<p style=\"text-align: center;\">[latex]C = \\frac{5}{9}(F - 32)[\/latex]<\/p>\r\nand substitutes [latex]75[\/latex] for [latex]F[\/latex] to calculate\r\n<p style=\"text-align: center;\">[latex]\\frac{5}{9}(75 - 32) \\approx 24^\\circ C[\/latex]<\/p>\r\nKnowing that a comfortable [latex]75[\/latex] degrees Fahrenheit is about [latex]24[\/latex] degrees Celsius, Betty gets the week's weather forecast for Milan, and wants to convert all of the temperatures to degrees Fahrenheit.\r\n\r\n<img class=\"size-full wp-image-3913 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/16115317\/3269ffa6058608835a4ec729a02a4d12b7c415d2.jpg\" alt=\"A forecast of Monday\u2019s through Thursday\u2019s weather.\" width=\"731\" height=\"226\" \/>\r\n\r\nAt first, Betty considers using the formula she has already found to complete the conversions. After all, she knows her algebra, and can easily solve the equation for [latex]F[\/latex] after substituting a value for [latex]C[\/latex]. For example, to convert [latex]26[\/latex] degrees Celsius, she could write:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} 26 &amp; = &amp; \\frac{5}{9}(F - 32) \\\\ 26 \\cdot \\frac{9}{5} &amp; = &amp; F - 32 \\\\ F &amp; = &amp; 26 \\cdot \\frac{9}{5} + 32 \\approx 79 \\end{array}[\/latex]<\/p>\r\n<p id=\"fs-id1165137540705\">After considering this option for a moment, however, she realizes that solving the equation for each of the temperatures will be awfully tedious. She realizes that since evaluation is easier than solving, it would be much more convenient to have a different formula, one that takes the Celsius temperature and outputs the Fahrenheit temperature.<\/p>\r\n<p id=\"fs-id1165137827441\">The formula for which Betty is searching corresponds to the idea of an\u00a0<strong>inverse function<\/strong>, which is a function for which the input of the original function becomes the output of the inverse function and the output of the original function becomes the input of the inverse function.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>inverse function<\/h3>\r\nAn <strong>inverse function<\/strong>, denoted as [latex]f^{-1}(x)[\/latex] reverses the operation of the original function [latex]f(x)[\/latex]. <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">The notation<\/span><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]f^{-1}[\/latex] is read \"<\/span><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]f[\/latex] inverse.\"<\/span>\r\n\r\nFor a function to have an inverse, it must be one-to-one (injective), meaning each output corresponds to exactly one input.\r\n\r\n&nbsp;\r\n\r\nProperties of an inverse function\r\n<ul>\r\n \t<li><strong>Symmetry:<\/strong> The graph of the inverse function is a reflection of the graph of the original function across the line [latex]y = x[\/latex]. If the point [latex](a,b)[\/latex] lies on the graph of [latex]f(x)[\/latex], then the point [latex](b,a)[\/latex] lies on the graph of [latex]f^{-1}(x)[\/latex].<\/li>\r\n \t<li><strong>Reversibility:<\/strong> The function and its inverse satisfy the conditions: [latex]f(f^{-1}(x)) = x[\/latex] and [latex]f^{-1}(f(x)) = x[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>Given a function [latex]f(x)[\/latex], we represent its inverse as [latex]f^{-1}(x)[\/latex], read as \"[latex]f[\/latex] inverse of [latex]x[\/latex].\"\r\n\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">The raised [latex]-1[\/latex] is part of the notation. It is not an exponent; it does not imply a power of [latex]-1[\/latex]. In other words, [latex]f^{-1}(x)[\/latex] does not mean [latex]\\frac{1}{f(x)}[\/latex] because [latex]\\frac{1}{f(x)}[\/latex] is the reciprocal of [latex]f[\/latex] and not the inverse.<\/section><section class=\"textbox example\" aria-label=\"Example\">Given a function [latex]f(x)[\/latex], we can verify whether some other function [latex]g(x)[\/latex] is the inverse of [latex]f(x)[\/latex] by checking if both [latex]g(f(x)) = x[\/latex] and [latex]f(g(x)) = x[\/latex] are true.\r\n[latex]\\\\[\/latex]\r\nFor example, [latex]y = 4x[\/latex] and [latex]y = \\frac{1}{4}x[\/latex] are inverse functions.\r\n<p style=\"text-align: center;\">[latex](f^{-1} \\circ f)(x) = f^{-1}(4x) = \\frac{1}{4}(4x) = x[\/latex]<\/p>\r\nand\r\n<p style=\"text-align: center;\">[latex](f \\circ f^{-1})(x) = f\\left(\\frac{1}{4}x\\right) = 4\\left(\\frac{1}{4}x\\right) = x[\/latex]<\/p>\r\nA few coordinate pairs from the graph of the function [latex]y = 4x[\/latex] are [latex](-2, -8)[\/latex], [latex](0, 0)[\/latex], and [latex](2, 8)[\/latex]. A few coordinate pairs from the graph of the function [latex]y = \\frac{1}{4}x[\/latex] are [latex](-8, -2)[\/latex], [latex](0, 0)[\/latex], and [latex](8, 2)[\/latex]. If we interchange the input and output of each coordinate pair of a function, the interchanged coordinate pairs would appear on the graph of the inverse function.\r\n\r\n<\/section>In simpler terms, inverse functions undo each other. If you graph both functions, the coordinates of one function\u2019s graph can be swapped to appear on the graph of its inverse.\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">A function is one-to-one if each output is paired with exactly one input. This means no two different inputs produce the same output. Mathematically, for a function [latex]f(x)[\/latex], if [latex]f(a)=f(b)[\/latex], then [latex]a=b[\/latex].\r\n[latex]\\\\[\/latex]\r\nA one-to-one function passes the horizontal line test\u2014any horizontal line drawn through the graph intersects the graph at most once.<\/section><section class=\"textbox example\" aria-label=\"Example\">If for a particular one-to-one function [latex]f\\left(2\\right)=4[\/latex] and [latex]f\\left(5\\right)=12[\/latex], what are the corresponding input and output values for the inverse function?[reveal-answer q=\"348517\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"348517\"]The inverse function reverses the input and output quantities, so if\r\n<p style=\"text-align: center;\">[latex]f\\left(2\\right)=4[\/latex], then [latex]{f}^{-1}\\left(4\\right)=2[\/latex]<\/p>\r\n<p style=\"text-align: left;\">and if<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(5\\right)=12[\/latex], then [latex]{f}^{-1}\\left(12\\right)=5[\/latex]<\/p>\r\nAlternatively, if we want to name the inverse function [latex]g[\/latex], then [latex]g\\left(4\\right)=2[\/latex] and [latex]g\\left(12\\right)=5[\/latex].\r\n\r\n<strong>Analysis of the Solution<\/strong>\r\n\r\nNotice that if we show the coordinate pairs in a table form, the input and output are clearly reversed.\r\n<table summary=\"For (x,f(x)) we have the values (2, 4) and (5, 12); for (x, g(x)), we have the values (4, 2) and (12, 5).\">\r\n<thead>\r\n<tr>\r\n<th>[latex]\\left(x,f\\left(x\\right)\\right)[\/latex]<\/th>\r\n<th>[latex]\\left(x,g\\left(x\\right)\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\left(2,4\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(4,2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\left(5,12\\right)[\/latex]<\/td>\r\n<td>[latex]\\left(12,5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given two functions [latex]f\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)[\/latex], test whether the functions are inverses of each other.<\/strong>\r\n<ol>\r\n \t<li>Determine whether [latex]f\\left(g\\left(x\\right)\\right)=x[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)=x[\/latex].<\/li>\r\n \t<li>If both statements are true, then [latex]g={f}^{-1}[\/latex] and [latex]f={g}^{-1}[\/latex]. If either statement is false, then [latex]g\\ne {f}^{-1}[\/latex] and [latex]f\\ne {g}^{-1}[\/latex].<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">If [latex]f\\left(x\\right)=\\dfrac{1}{x+2}[\/latex] and [latex]g\\left(x\\right)=\\dfrac{1}{x}-2[\/latex], is [latex]g={f}^{-1}?[\/latex][reveal-answer q=\"421291\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"421291\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{align} g\\left(f\\left(x\\right)\\right)&amp;=\\frac{1}{\\left(\\frac{1}{x+2}\\right)}{-2 }\\\\[1.5mm]&amp;={ x }+{ 2 } -{ 2 }\\\\[1.5mm]&amp;={ x } \\end{align}[\/latex]<\/p>\r\nso\r\n<p style=\"text-align: center;\">[latex]g={f}^{-1}\\text{ and }f={g}^{-1}[\/latex]<\/p>\r\nThis is enough to answer yes to the question, but we can also verify the other formula.\r\n<p style=\"text-align: center;\">[latex]\\begin{align} f\\left(g\\left(x\\right)\\right)&amp;=\\frac{1}{\\frac{1}{x}-2+2}\\\\[1.5mm] &amp;=\\frac{1}{\\frac{1}{x}} \\\\[1.5mm] &amp;=x \\end{align}[\/latex]<\/p>\r\n<strong>Analysis of the Solution<\/strong>\r\n\r\nNotice the inverse operations are in reverse order of the operations from the original function.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">If [latex]f\\left(x\\right)={x}^{3}[\/latex] (the cube function) and [latex]g\\left(x\\right)=\\frac{1}{3}x[\/latex], is [latex]g={f}^{-1}?[\/latex][reveal-answer q=\"637419\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"637419\"][latex]f\\left(g\\left(x\\right)\\right)=\\left(\\frac{1}{3}x\\right)^3=\\dfrac{{x}^{3}}{27}\\ne x[\/latex]No, the functions are not inverses.\r\n[latex]\\\\[\/latex]\r\n<strong>Analysis of the Solution\r\n<\/strong>[latex]\\\\[\/latex]<strong>\r\n<\/strong>The correct inverse to [latex]x^3[\/latex] is the cube root [latex]\\sqrt[3]{x}={x}^{\\frac{1}{3}}[\/latex], that is, the one-third is an exponent, not a multiplier.[\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]317938[\/ohm_question]<\/section><\/div>\r\n<section id=\"fs-id1165137605437\">\r\n<div id=\"fs-id1165137911739\" class=\"solution\"><\/div>\r\n<\/section>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Use composition to check if two functions are inverses<\/li>\n<li>Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one<\/li>\n<li>Find or calculate the inverse of a function<\/li>\n<li>Use the graph of a function to draw its inverse<\/li>\n<\/ul>\n<\/section>\n<h2>Inverse Function<\/h2>\n<p>Betty is traveling to Milan for a fashion show and wants to know what the temperature will be. She is not familiar with the Celsius scale. To get an idea of how temperature measurements are related, Betty wants to convert 75 degrees Fahrenheit to degrees Celsius using the formula<\/p>\n<p style=\"text-align: center;\">[latex]C = \\frac{5}{9}(F - 32)[\/latex]<\/p>\n<p>and substitutes [latex]75[\/latex] for [latex]F[\/latex] to calculate<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{5}{9}(75 - 32) \\approx 24^\\circ C[\/latex]<\/p>\n<p>Knowing that a comfortable [latex]75[\/latex] degrees Fahrenheit is about [latex]24[\/latex] degrees Celsius, Betty gets the week&#8217;s weather forecast for Milan, and wants to convert all of the temperatures to degrees Fahrenheit.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-3913 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/05\/16115317\/3269ffa6058608835a4ec729a02a4d12b7c415d2.jpg\" alt=\"A forecast of Monday\u2019s through Thursday\u2019s weather.\" width=\"731\" height=\"226\" \/><\/p>\n<p>At first, Betty considers using the formula she has already found to complete the conversions. After all, she knows her algebra, and can easily solve the equation for [latex]F[\/latex] after substituting a value for [latex]C[\/latex]. For example, to convert [latex]26[\/latex] degrees Celsius, she could write:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rcl} 26 & = & \\frac{5}{9}(F - 32) \\\\ 26 \\cdot \\frac{9}{5} & = & F - 32 \\\\ F & = & 26 \\cdot \\frac{9}{5} + 32 \\approx 79 \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165137540705\">After considering this option for a moment, however, she realizes that solving the equation for each of the temperatures will be awfully tedious. She realizes that since evaluation is easier than solving, it would be much more convenient to have a different formula, one that takes the Celsius temperature and outputs the Fahrenheit temperature.<\/p>\n<p id=\"fs-id1165137827441\">The formula for which Betty is searching corresponds to the idea of an\u00a0<strong>inverse function<\/strong>, which is a function for which the input of the original function becomes the output of the inverse function and the output of the original function becomes the input of the inverse function.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>inverse function<\/h3>\n<p>An <strong>inverse function<\/strong>, denoted as [latex]f^{-1}(x)[\/latex] reverses the operation of the original function [latex]f(x)[\/latex]. <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">The notation<\/span><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]f^{-1}[\/latex] is read &#8220;<\/span><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]f[\/latex] inverse.&#8221;<\/span><\/p>\n<p>For a function to have an inverse, it must be one-to-one (injective), meaning each output corresponds to exactly one input.<\/p>\n<p>&nbsp;<\/p>\n<p>Properties of an inverse function<\/p>\n<ul>\n<li><strong>Symmetry:<\/strong> The graph of the inverse function is a reflection of the graph of the original function across the line [latex]y = x[\/latex]. If the point [latex](a,b)[\/latex] lies on the graph of [latex]f(x)[\/latex], then the point [latex](b,a)[\/latex] lies on the graph of [latex]f^{-1}(x)[\/latex].<\/li>\n<li><strong>Reversibility:<\/strong> The function and its inverse satisfy the conditions: [latex]f(f^{-1}(x)) = x[\/latex] and [latex]f^{-1}(f(x)) = x[\/latex]<\/li>\n<\/ul>\n<\/section>\n<p>Given a function [latex]f(x)[\/latex], we represent its inverse as [latex]f^{-1}(x)[\/latex], read as &#8220;[latex]f[\/latex] inverse of [latex]x[\/latex].&#8221;<\/p>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">The raised [latex]-1[\/latex] is part of the notation. It is not an exponent; it does not imply a power of [latex]-1[\/latex]. In other words, [latex]f^{-1}(x)[\/latex] does not mean [latex]\\frac{1}{f(x)}[\/latex] because [latex]\\frac{1}{f(x)}[\/latex] is the reciprocal of [latex]f[\/latex] and not the inverse.<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Given a function [latex]f(x)[\/latex], we can verify whether some other function [latex]g(x)[\/latex] is the inverse of [latex]f(x)[\/latex] by checking if both [latex]g(f(x)) = x[\/latex] and [latex]f(g(x)) = x[\/latex] are true.<br \/>\n[latex]\\\\[\/latex]<br \/>\nFor example, [latex]y = 4x[\/latex] and [latex]y = \\frac{1}{4}x[\/latex] are inverse functions.<\/p>\n<p style=\"text-align: center;\">[latex](f^{-1} \\circ f)(x) = f^{-1}(4x) = \\frac{1}{4}(4x) = x[\/latex]<\/p>\n<p>and<\/p>\n<p style=\"text-align: center;\">[latex](f \\circ f^{-1})(x) = f\\left(\\frac{1}{4}x\\right) = 4\\left(\\frac{1}{4}x\\right) = x[\/latex]<\/p>\n<p>A few coordinate pairs from the graph of the function [latex]y = 4x[\/latex] are [latex](-2, -8)[\/latex], [latex](0, 0)[\/latex], and [latex](2, 8)[\/latex]. A few coordinate pairs from the graph of the function [latex]y = \\frac{1}{4}x[\/latex] are [latex](-8, -2)[\/latex], [latex](0, 0)[\/latex], and [latex](8, 2)[\/latex]. If we interchange the input and output of each coordinate pair of a function, the interchanged coordinate pairs would appear on the graph of the inverse function.<\/p>\n<\/section>\n<p>In simpler terms, inverse functions undo each other. If you graph both functions, the coordinates of one function\u2019s graph can be swapped to appear on the graph of its inverse.<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">A function is one-to-one if each output is paired with exactly one input. This means no two different inputs produce the same output. Mathematically, for a function [latex]f(x)[\/latex], if [latex]f(a)=f(b)[\/latex], then [latex]a=b[\/latex].<br \/>\n[latex]\\\\[\/latex]<br \/>\nA one-to-one function passes the horizontal line test\u2014any horizontal line drawn through the graph intersects the graph at most once.<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">If for a particular one-to-one function [latex]f\\left(2\\right)=4[\/latex] and [latex]f\\left(5\\right)=12[\/latex], what are the corresponding input and output values for the inverse function?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q348517\">Show Solution<\/button><\/p>\n<div id=\"q348517\" class=\"hidden-answer\" style=\"display: none\">The inverse function reverses the input and output quantities, so if<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(2\\right)=4[\/latex], then [latex]{f}^{-1}\\left(4\\right)=2[\/latex]<\/p>\n<p style=\"text-align: left;\">and if<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(5\\right)=12[\/latex], then [latex]{f}^{-1}\\left(12\\right)=5[\/latex]<\/p>\n<p>Alternatively, if we want to name the inverse function [latex]g[\/latex], then [latex]g\\left(4\\right)=2[\/latex] and [latex]g\\left(12\\right)=5[\/latex].<\/p>\n<p><strong>Analysis of the Solution<\/strong><\/p>\n<p>Notice that if we show the coordinate pairs in a table form, the input and output are clearly reversed.<\/p>\n<table summary=\"For (x,f(x)) we have the values (2, 4) and (5, 12); for (x, g(x)), we have the values (4, 2) and (12, 5).\">\n<thead>\n<tr>\n<th>[latex]\\left(x,f\\left(x\\right)\\right)[\/latex]<\/th>\n<th>[latex]\\left(x,g\\left(x\\right)\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\left(2,4\\right)[\/latex]<\/td>\n<td>[latex]\\left(4,2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\left(5,12\\right)[\/latex]<\/td>\n<td>[latex]\\left(12,5\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given two functions [latex]f\\left(x\\right)[\/latex] and [latex]g\\left(x\\right)[\/latex], test whether the functions are inverses of each other.<\/strong><\/p>\n<ol>\n<li>Determine whether [latex]f\\left(g\\left(x\\right)\\right)=x[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)=x[\/latex].<\/li>\n<li>If both statements are true, then [latex]g={f}^{-1}[\/latex] and [latex]f={g}^{-1}[\/latex]. If either statement is false, then [latex]g\\ne {f}^{-1}[\/latex] and [latex]f\\ne {g}^{-1}[\/latex].<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">If [latex]f\\left(x\\right)=\\dfrac{1}{x+2}[\/latex] and [latex]g\\left(x\\right)=\\dfrac{1}{x}-2[\/latex], is [latex]g={f}^{-1}?[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q421291\">Show Solution<\/button><\/p>\n<div id=\"q421291\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align} g\\left(f\\left(x\\right)\\right)&=\\frac{1}{\\left(\\frac{1}{x+2}\\right)}{-2 }\\\\[1.5mm]&={ x }+{ 2 } -{ 2 }\\\\[1.5mm]&={ x } \\end{align}[\/latex]<\/p>\n<p>so<\/p>\n<p style=\"text-align: center;\">[latex]g={f}^{-1}\\text{ and }f={g}^{-1}[\/latex]<\/p>\n<p>This is enough to answer yes to the question, but we can also verify the other formula.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} f\\left(g\\left(x\\right)\\right)&=\\frac{1}{\\frac{1}{x}-2+2}\\\\[1.5mm] &=\\frac{1}{\\frac{1}{x}} \\\\[1.5mm] &=x \\end{align}[\/latex]<\/p>\n<p><strong>Analysis of the Solution<\/strong><\/p>\n<p>Notice the inverse operations are in reverse order of the operations from the original function.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">If [latex]f\\left(x\\right)={x}^{3}[\/latex] (the cube function) and [latex]g\\left(x\\right)=\\frac{1}{3}x[\/latex], is [latex]g={f}^{-1}?[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q637419\">Show Solution<\/button><\/p>\n<div id=\"q637419\" class=\"hidden-answer\" style=\"display: none\">[latex]f\\left(g\\left(x\\right)\\right)=\\left(\\frac{1}{3}x\\right)^3=\\dfrac{{x}^{3}}{27}\\ne x[\/latex]No, the functions are not inverses.<br \/>\n[latex]\\\\[\/latex]<br \/>\n<strong>Analysis of the Solution<br \/>\n<\/strong>[latex]\\\\[\/latex]<strong><br \/>\n<\/strong>The correct inverse to [latex]x^3[\/latex] is the cube root [latex]\\sqrt[3]{x}={x}^{\\frac{1}{3}}[\/latex], that is, the one-third is an exponent, not a multiplier.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm317938\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317938&theme=lumen&iframe_resize_id=ohm317938&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<section id=\"fs-id1165137605437\">\n<div id=\"fs-id1165137911739\" class=\"solution\"><\/div>\n<\/section>\n","protected":false},"author":6,"menu_order":22,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax 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