{"id":5115,"date":"2018-01-02T00:03:28","date_gmt":"2018-01-02T00:03:28","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/?post_type=chapter&#038;p=5115"},"modified":"2023-11-08T13:20:54","modified_gmt":"2023-11-08T13:20:54","slug":"composite-and-inverse-functions","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/chapter\/composite-and-inverse-functions\/","title":{"raw":"12.2 - Composite and Inverse Functions","rendered":"12.2 &#8211; Composite and Inverse Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>(12.2.1) - Define a composite function<\/li>\r\n \t<li>(12.2.2) - Define an inverse function\r\n<ul>\r\n \t<li>Use compositions of functions to verify inverses algebraically<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h1>(12.2.1) - Define a composite function<\/h1>\r\nSuppose we want to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day of the year. Notice how we have just defined two relationships: The cost depends on the temperature, and the temperature depends on the day.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200743\/CNX_Precalc_Figure_01_04_0062.jpg\" alt=\"Explanation of C(T(5)), which is the cost for the temperature and T(5) is the temperature on day 5.\" width=\"487\" height=\"140\" \/> <b>Figure 1<\/b>[\/caption]\r\n<p id=\"fs-id1165134038788\">Using descriptive variables, we can notate these two functions. The function [latex]C\\left(T\\right)[\/latex] gives the cost [latex]C[\/latex] of heating a house for a given average daily temperature in [latex]T[\/latex] degrees Celsius. The function [latex]T\\left(d\\right)[\/latex] gives the average daily temperature on day [latex]d[\/latex] of the year. For any given day, [latex]\\text{Cost}=C\\left(T\\left(d\\right)\\right)[\/latex] means that the cost depends on the temperature, which in turns depends on the day of the year. Thus, we can evaluate the cost function at the temperature [latex]T\\left(d\\right)[\/latex]. For example, we could evaluate [latex]T\\left(5\\right)[\/latex] to determine the average daily temperature on the 5th day of the year. Then, we could evaluate the <strong>cost function<\/strong> at that temperature. We would write [latex]C\\left(T\\left(5\\right)\\right)[\/latex].\u00a0By combining these two relationships into one function, we have performed function composition.<\/p>\r\nWe read the left-hand side as [latex]\"f[\/latex] composed with [latex]g[\/latex] at [latex]x,\"[\/latex] and the right-hand side as [latex]\"f[\/latex] of [latex]g[\/latex] of [latex]x.\"[\/latex] The two sides of the equation have the same mathematical meaning and are equal. The open circle symbol [latex]\\circ [\/latex] is called the composition operator.\r\n\r\nIt is also important to understand the order of operations in evaluating a composite function. We follow the usual convention with parentheses by starting with the innermost parentheses first, and then working to the outside.\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200744\/CNX_Precalc_Figure_01_04_0012.jpg\" alt=\"Explanation of the composite function. g(x), the output of g is the input of f. X is the input of g.\" width=\"487\" height=\"171\" \/>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUsing the functions provided, find [latex]f\\left(g\\left(x\\right)\\right)[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)[\/latex].\r\n\r\n[latex]f\\left(x\\right)=2x+1[\/latex]\r\n\r\n[latex]g\\left(x\\right)=3-x[\/latex]\r\n[reveal-answer q=\"337338\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"337338\"]\r\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=2x+1[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]g\\left(x\\right)=3-x[\/latex]<\/p>\r\nLet\u2019s begin by substituting [latex]g\\left(x\\right)[\/latex] into [latex]f\\left(x\\right)[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}f\\left(g\\left(x\\right)\\right)&amp;=&amp;2\\left(3-x\\right)+1\\hfill \\\\ \\text{ }&amp;=&amp;6 - 2x+1\\hfill \\\\ \\text{ }&amp;=&amp;7 - 2x\\hfill \\end{array}[\/latex]<\/p>\r\nNow we can substitute [latex]f\\left(x\\right)[\/latex] into [latex]g\\left(x\\right)[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}g\\left(f\\left(x\\right)\\right)&amp;=&amp;3-\\left(2x+1\\right)\\hfill \\\\ \\text{ }&amp;=&amp;3 - 2x - 1\\hfill \\\\ \\text{ }&amp;=&amp;-2x+2\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video you will see another example of how to find the composition of two functions.\r\n\r\nhttps:\/\/youtu.be\/r_LssVS4NHk\r\n<h1>(12.2.2) - Define an inverse function<\/h1>\r\n<p id=\"fs-id1165137827441\">An <strong>inverse function<\/strong>\u00a0is a function for which the input of the original function becomes the output of the inverse function. This naturally leads to the output of the original function becoming the input of the inverse function. The reason we want to introduce inverse functions is because exponential and logarithmic functions are inverses of each other, and understanding this quality helps to make understanding logarithmic functions easier. And the reason we introduced composite functions is because you can verify, algebraically, whether two functions are inverses of each other by using a composition.<\/p>\r\n<p id=\"fs-id1165135528385\">Given a function [latex]f\\left(x\\right)[\/latex], we represent its inverse as [latex]{f}^{-1}\\left(x\\right)[\/latex], read as [latex]\"f[\/latex] inverse of [latex]x.\"[\/latex] 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}\\left(x\\right)[\/latex] does <em>not<\/em> mean [latex]\\displaystyle \\frac{1}{f\\left(x\\right)}[\/latex] because [latex]\\displaystyle \\frac{1}{f\\left(x\\right)}[\/latex] is the reciprocal of [latex]f[\/latex] and not the inverse.<\/p>\r\n<p id=\"fs-id1165137724926\">Just as zero does not have a <strong>reciprocal<\/strong>, some functions do not have inverses.<\/p>\r\n\r\n<div id=\"fs-id1165137933105\" class=\"note textbox\">\r\n<h3 class=\"title\">Inverse Function<\/h3>\r\n<p id=\"fs-id1165137473076\">For any\u00a0function [latex]f\\left(x\\right)[\/latex], a function [latex]{f}^{-1}\\left(x\\right)[\/latex] is an <strong>inverse function<\/strong>\u00a0if [latex]f[\/latex] if [latex](f \\circ {f}^{-1})(x) = x[\/latex] and\u00a0[latex](f^{-1} \\circ f)(x) = x[\/latex]<\/p>\r\n<p id=\"fs-id1165137444821\">The notation [latex]{f}^{-1}[\/latex] is read [latex]\\text{\"}f[\/latex] inverse.\" Like any other function, we can use any variable name as the input for [latex]{f}^{-1}[\/latex], so we will often write [latex]{f}^{-1}\\left(x\\right)[\/latex], which we read as [latex]\"f[\/latex] inverse of [latex]x.\"[\/latex]\r\nKeep in mind that<\/p>\r\n\r\n<div id=\"fs-id1165137581324\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\displaystyle {f}^{-1}\\left(x\\right)\\ne \\frac{1}{f\\left(x\\right)}[\/latex]<\/div>\r\n<p id=\"fs-id1165135194095\">and not all functions have inverses.<\/p>\r\n\r\n<\/div>\r\nIn our first example we will identify an inverse function from ordered pairs.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIf for a particular function, which has an inverse, [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?\r\n[reveal-answer q=\"664782\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"664782\"]\r\n<p id=\"fs-id1165137737081\">The inverse function reverses the input and output quantities, so if<\/p>\r\n\r\n<div id=\"fs-id1165137462459\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{array}{c}f\\left(2\\right)=4,\\text{ then }{f}^{-1}\\left(4\\right)=2;\\\\ f\\left(5\\right)=12,{\\text{ then } f}^{-1}\\left(12\\right)=5.\\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137659464\">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>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<div id=\"Example_01_07_01\" class=\"example\">\r\n<div id=\"fs-id1165137656641\" class=\"exercise\">\r\n<div id=\"fs-id1165135245520\" class=\"commentary\">\r\n<p id=\"fs-id1165135508518\">Notice that if we show the coordinate pairs in a table form, the input and output are clearly reversed.<\/p>\r\n\r\n<table style=\"width: 30%\" 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 style=\"width: 49.345%\">[latex]\\left(x,f\\left(x\\right)\\right)[\/latex]<\/th>\r\n<th style=\"width: 2.18341%\">[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 style=\"width: 49.345%\">[latex]\\left(2,4\\right)[\/latex]<\/td>\r\n<td style=\"width: 2.18341%\">[latex]\\left(4,2\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 49.345%\">[latex]\\left(5,12\\right)[\/latex]<\/td>\r\n<td style=\"width: 2.18341%\">[latex]\\left(12,5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"Example_01_07_01\" class=\"example\">\r\n<div id=\"fs-id1165137656641\" class=\"exercise\">\r\n<div id=\"fs-id1165135245520\" class=\"commentary\">In the following video we show an example of finding corresponding input and output values given two ordered pairs from functions that are inverses.<\/div>\r\n<\/div>\r\n<\/div>\r\nhttps:\/\/youtu.be\/IR_1L1mnpvw\r\n<div id=\"fs-id1165134357354\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135434077\">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.<\/h3>\r\n<ol id=\"fs-id1165137452358\">\r\n \t<li>Substitute\u00a0[latex]g(x)[\/latex]\u00a0into [latex]f(x)[\/latex]. The result must be [latex]x[\/latex]. [latex]f\\left(g(x)\\right)=x[\/latex]<\/li>\r\n \t<li>Substitute\u00a0[latex]f(x)[\/latex]\u00a0into [latex]g(x)[\/latex]. The result must be [latex]x[\/latex]. [latex]g\\left(f(x)\\right)=x[\/latex]<\/li>\r\n<\/ol>\r\n<p style=\"text-align: center\">If\u00a0[latex]f(x)[\/latex] and\u00a0\u00a0[latex]g(x)[\/latex] are inverses, then\u00a0\u00a0[latex]f(x)=g^{-1}(x)[\/latex] and\u00a0[latex]g(x)=f^{-1}(x)[\/latex]<\/p>\r\n\r\n<\/div>\r\nIn our next example we will test inverse relationships algebraically.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIf [latex]f\\left(x\\right)=x^2-3[\/latex], for [latex]x\\ge0[\/latex] and [latex]g\\left(x\\right)=\\sqrt{x+3}[\/latex], are [latex]g[\/latex] and [latex]f[\/latex] inverses of each other?\r\n[reveal-answer q=\"598434\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"598434\"]\r\n\r\nSubstitute [latex]g(x)=\\sqrt{x+3}[\/latex] into [latex]f(x)[\/latex], this means the new variable in\u00a0[latex]f(x)[\/latex] is [latex]\\sqrt{x+3}[\/latex], so we will substitute that expression where we see [latex]x[\/latex]. \u00a0Using parentheses helps keep track of things.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}f\\left(\\sqrt{x+3}\\right)&amp;=&amp;{(\\sqrt{x+3})}^2-3\\hfill\\\\&amp;=&amp;x+3-3\\\\&amp;=&amp;x\\hfill \\end{array}[\/latex]<\/p>\r\nNow, we substitute [latex]f(x) = x^2-3[\/latex] into [latex]g(x)[\/latex]. this means the new variable in [latex]g(x)[\/latex] is [latex]x^2-3[\/latex], so we will substitute that expression where we see [latex]x[\/latex]:\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}g\\left(x^2-3\\right)&amp;=&amp; \\sqrt{\\left(x^2-3\\right)+3}\\\\&amp;=&amp;\\sqrt{\\left(x^2\\right)}\\\\&amp;=&amp;x\\hfill \\end{array}[\/latex]<\/p>\r\nOur result implies that [latex]g(x)[\/latex] and\u00a0 [latex]f(x)[\/latex] are inverses of each other.\r\n<h4>Answer<\/h4>\r\nYes, [latex]f(x)[\/latex] and [latex]g(x)[\/latex] are inverses of each other,\u00a0for [latex]x\\ge0[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we use algebra to determine if two functions are inverses.\r\n\r\nhttps:\/\/youtu.be\/vObCvTOatfQ\r\n\r\nWe will show another example of how to verify whether you have an inverse algebraically.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIf [latex]\\displaystyle f\\left(x\\right)=\\frac{1}{x+2}[\/latex] and [latex]\\displaystyle g\\left(x\\right)=\\frac{1}{x}-2[\/latex], are [latex]g[\/latex] and [latex]f[\/latex] inverses of each other?\r\n[reveal-answer q=\"56557\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"56557\"]\r\n\r\nSubstitute [latex]\\displaystyle g(x)=\\frac{1}{x}-2[\/latex] into [latex]f(x)[\/latex], this means the new variable in\u00a0[latex]f(x)[\/latex] is [latex]\\displaystyle \\frac{1}{x}-2[\/latex] so you will substitute that expression where you see [latex]x[\/latex]. \u00a0Using parentheses helps keep track of things.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{cc} f\\left(\\displaystyle \\frac{1}{x}-2\\right)&amp;=&amp;\\displaystyle \\frac{1}{\\left(\\frac{1}{x}-2\\right)+2}\\\\&amp;=&amp;\\displaystyle \\frac{1}{\\frac{1}{x}}\\\\&amp;=&amp;{ x }\\hfill \\end{array}[\/latex]<\/p>\r\nNext, we substitute [latex]\\displaystyle f(x)=\\frac{1}{x+2}[\/latex] into [latex]g(x)[\/latex], this means the new variable in\u00a0[latex]g(x)[\/latex] is [latex]\\displaystyle \\frac{1}{x+2}[\/latex] so you will substitute that expression where you see [latex]x[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{cc} g \\left(\\displaystyle \\frac{1}{x+2}\\right) &amp;=&amp;\\displaystyle \\frac{1}{\\left(\\frac{1}{x+2}\\right)}-2 \\\\ &amp;=&amp; x+2 -2 \\\\ &amp;=&amp; x \\hfill\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nYes, [latex]f[\/latex] and [latex]g[\/latex] are inverse functions.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe will show one more example of how to use algebra to determine whether two functions are inverses of each other.\r\n\r\nhttps:\/\/www.youtube.com\/watch?v=hzehBtNmw08&amp;feature=youtu.be","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>(12.2.1) &#8211; Define a composite function<\/li>\n<li>(12.2.2) &#8211; Define an inverse function\n<ul>\n<li>Use compositions of functions to verify inverses algebraically<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h1>(12.2.1) &#8211; Define a composite function<\/h1>\n<p>Suppose we want to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day of the year. Notice how we have just defined two relationships: The cost depends on the temperature, and the temperature depends on the day.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200743\/CNX_Precalc_Figure_01_04_0062.jpg\" alt=\"Explanation of C(T(5)), which is the cost for the temperature and T(5) is the temperature on day 5.\" width=\"487\" height=\"140\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165134038788\">Using descriptive variables, we can notate these two functions. The function [latex]C\\left(T\\right)[\/latex] gives the cost [latex]C[\/latex] of heating a house for a given average daily temperature in [latex]T[\/latex] degrees Celsius. The function [latex]T\\left(d\\right)[\/latex] gives the average daily temperature on day [latex]d[\/latex] of the year. For any given day, [latex]\\text{Cost}=C\\left(T\\left(d\\right)\\right)[\/latex] means that the cost depends on the temperature, which in turns depends on the day of the year. Thus, we can evaluate the cost function at the temperature [latex]T\\left(d\\right)[\/latex]. For example, we could evaluate [latex]T\\left(5\\right)[\/latex] to determine the average daily temperature on the 5th day of the year. Then, we could evaluate the <strong>cost function<\/strong> at that temperature. We would write [latex]C\\left(T\\left(5\\right)\\right)[\/latex].\u00a0By combining these two relationships into one function, we have performed function composition.<\/p>\n<p>We read the left-hand side as [latex]\"f[\/latex] composed with [latex]g[\/latex] at [latex]x,\"[\/latex] and the right-hand side as [latex]\"f[\/latex] of [latex]g[\/latex] of [latex]x.\"[\/latex] The two sides of the equation have the same mathematical meaning and are equal. The open circle symbol [latex]\\circ[\/latex] is called the composition operator.<\/p>\n<p>It is also important to understand the order of operations in evaluating a composite function. We follow the usual convention with parentheses by starting with the innermost parentheses first, and then working to the outside.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200744\/CNX_Precalc_Figure_01_04_0012.jpg\" alt=\"Explanation of the composite function. g(x), the output of g is the input of f. X is the input of g.\" width=\"487\" height=\"171\" \/><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Using the functions provided, find [latex]f\\left(g\\left(x\\right)\\right)[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)[\/latex].<\/p>\n<p>[latex]f\\left(x\\right)=2x+1[\/latex]<\/p>\n<p>[latex]g\\left(x\\right)=3-x[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q337338\">Show Answer<\/span><\/p>\n<div id=\"q337338\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=2x+1[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]g\\left(x\\right)=3-x[\/latex]<\/p>\n<p>Let\u2019s begin by substituting [latex]g\\left(x\\right)[\/latex] into [latex]f\\left(x\\right)[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}f\\left(g\\left(x\\right)\\right)&=&2\\left(3-x\\right)+1\\hfill \\\\ \\text{ }&=&6 - 2x+1\\hfill \\\\ \\text{ }&=&7 - 2x\\hfill \\end{array}[\/latex]<\/p>\n<p>Now we can substitute [latex]f\\left(x\\right)[\/latex] into [latex]g\\left(x\\right)[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}g\\left(f\\left(x\\right)\\right)&=&3-\\left(2x+1\\right)\\hfill \\\\ \\text{ }&=&3 - 2x - 1\\hfill \\\\ \\text{ }&=&-2x+2\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video you will see another example of how to find the composition of two functions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  Composition of Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/r_LssVS4NHk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h1>(12.2.2) &#8211; Define an inverse function<\/h1>\n<p id=\"fs-id1165137827441\">An <strong>inverse function<\/strong>\u00a0is a function for which the input of the original function becomes the output of the inverse function. This naturally leads to the output of the original function becoming the input of the inverse function. The reason we want to introduce inverse functions is because exponential and logarithmic functions are inverses of each other, and understanding this quality helps to make understanding logarithmic functions easier. And the reason we introduced composite functions is because you can verify, algebraically, whether two functions are inverses of each other by using a composition.<\/p>\n<p id=\"fs-id1165135528385\">Given a function [latex]f\\left(x\\right)[\/latex], we represent its inverse as [latex]{f}^{-1}\\left(x\\right)[\/latex], read as [latex]\"f[\/latex] inverse of [latex]x.\"[\/latex] 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}\\left(x\\right)[\/latex] does <em>not<\/em> mean [latex]\\displaystyle \\frac{1}{f\\left(x\\right)}[\/latex] because [latex]\\displaystyle \\frac{1}{f\\left(x\\right)}[\/latex] is the reciprocal of [latex]f[\/latex] and not the inverse.<\/p>\n<p id=\"fs-id1165137724926\">Just as zero does not have a <strong>reciprocal<\/strong>, some functions do not have inverses.<\/p>\n<div id=\"fs-id1165137933105\" class=\"note textbox\">\n<h3 class=\"title\">Inverse Function<\/h3>\n<p id=\"fs-id1165137473076\">For any\u00a0function [latex]f\\left(x\\right)[\/latex], a function [latex]{f}^{-1}\\left(x\\right)[\/latex] is an <strong>inverse function<\/strong>\u00a0if [latex]f[\/latex] if [latex](f \\circ {f}^{-1})(x) = x[\/latex] and\u00a0[latex](f^{-1} \\circ f)(x) = x[\/latex]<\/p>\n<p id=\"fs-id1165137444821\">The notation [latex]{f}^{-1}[\/latex] is read [latex]\\text{\"}f[\/latex] inverse.&#8221; Like any other function, we can use any variable name as the input for [latex]{f}^{-1}[\/latex], so we will often write [latex]{f}^{-1}\\left(x\\right)[\/latex], which we read as [latex]\"f[\/latex] inverse of [latex]x.\"[\/latex]<br \/>\nKeep in mind that<\/p>\n<div id=\"fs-id1165137581324\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\displaystyle {f}^{-1}\\left(x\\right)\\ne \\frac{1}{f\\left(x\\right)}[\/latex]<\/div>\n<p id=\"fs-id1165135194095\">and not all functions have inverses.<\/p>\n<\/div>\n<p>In our first example we will identify an inverse function from ordered pairs.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>If for a particular function, which has an inverse, [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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q664782\">Show Answer<\/span><\/p>\n<div id=\"q664782\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137737081\">The inverse function reverses the input and output quantities, so if<\/p>\n<div id=\"fs-id1165137462459\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{array}{c}f\\left(2\\right)=4,\\text{ then }{f}^{-1}\\left(4\\right)=2;\\\\ f\\left(5\\right)=12,{\\text{ then } f}^{-1}\\left(12\\right)=5.\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137659464\">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<h4>Analysis of the Solution<\/h4>\n<div id=\"Example_01_07_01\" class=\"example\">\n<div id=\"fs-id1165137656641\" class=\"exercise\">\n<div id=\"fs-id1165135245520\" class=\"commentary\">\n<p id=\"fs-id1165135508518\">Notice that if we show the coordinate pairs in a table form, the input and output are clearly reversed.<\/p>\n<table style=\"width: 30%\" 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 style=\"width: 49.345%\">[latex]\\left(x,f\\left(x\\right)\\right)[\/latex]<\/th>\n<th style=\"width: 2.18341%\">[latex]\\left(x,g\\left(x\\right)\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 49.345%\">[latex]\\left(2,4\\right)[\/latex]<\/td>\n<td style=\"width: 2.18341%\">[latex]\\left(4,2\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 49.345%\">[latex]\\left(5,12\\right)[\/latex]<\/td>\n<td style=\"width: 2.18341%\">[latex]\\left(12,5\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_01_07_01\" class=\"example\">\n<div id=\"fs-id1165137656641\" class=\"exercise\">\n<div id=\"fs-id1165135245520\" class=\"commentary\">In the following video we show an example of finding corresponding input and output values given two ordered pairs from functions that are inverses.<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Function and Inverse Function Values\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/IR_1L1mnpvw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div id=\"fs-id1165134357354\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135434077\">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.<\/h3>\n<ol id=\"fs-id1165137452358\">\n<li>Substitute\u00a0[latex]g(x)[\/latex]\u00a0into [latex]f(x)[\/latex]. The result must be [latex]x[\/latex]. [latex]f\\left(g(x)\\right)=x[\/latex]<\/li>\n<li>Substitute\u00a0[latex]f(x)[\/latex]\u00a0into [latex]g(x)[\/latex]. The result must be [latex]x[\/latex]. [latex]g\\left(f(x)\\right)=x[\/latex]<\/li>\n<\/ol>\n<p style=\"text-align: center\">If\u00a0[latex]f(x)[\/latex] and\u00a0\u00a0[latex]g(x)[\/latex] are inverses, then\u00a0\u00a0[latex]f(x)=g^{-1}(x)[\/latex] and\u00a0[latex]g(x)=f^{-1}(x)[\/latex]<\/p>\n<\/div>\n<p>In our next example we will test inverse relationships algebraically.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>If [latex]f\\left(x\\right)=x^2-3[\/latex], for [latex]x\\ge0[\/latex] and [latex]g\\left(x\\right)=\\sqrt{x+3}[\/latex], are [latex]g[\/latex] and [latex]f[\/latex] inverses of each other?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q598434\">Show Answer<\/span><\/p>\n<div id=\"q598434\" class=\"hidden-answer\" style=\"display: none\">\n<p>Substitute [latex]g(x)=\\sqrt{x+3}[\/latex] into [latex]f(x)[\/latex], this means the new variable in\u00a0[latex]f(x)[\/latex] is [latex]\\sqrt{x+3}[\/latex], so we will substitute that expression where we see [latex]x[\/latex]. \u00a0Using parentheses helps keep track of things.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}f\\left(\\sqrt{x+3}\\right)&=&{(\\sqrt{x+3})}^2-3\\hfill\\\\&=&x+3-3\\\\&=&x\\hfill \\end{array}[\/latex]<\/p>\n<p>Now, we substitute [latex]f(x) = x^2-3[\/latex] into [latex]g(x)[\/latex]. this means the new variable in [latex]g(x)[\/latex] is [latex]x^2-3[\/latex], so we will substitute that expression where we see [latex]x[\/latex]:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}g\\left(x^2-3\\right)&=& \\sqrt{\\left(x^2-3\\right)+3}\\\\&=&\\sqrt{\\left(x^2\\right)}\\\\&=&x\\hfill \\end{array}[\/latex]<\/p>\n<p>Our result implies that [latex]g(x)[\/latex] and\u00a0 [latex]f(x)[\/latex] are inverses of each other.<\/p>\n<h4>Answer<\/h4>\n<p>Yes, [latex]f(x)[\/latex] and [latex]g(x)[\/latex] are inverses of each other,\u00a0for [latex]x\\ge0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we use algebra to determine if two functions are inverses.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex 1: Determine if Two Functions Are Inverses\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vObCvTOatfQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>We will show another example of how to verify whether you have an inverse algebraically.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>If [latex]\\displaystyle f\\left(x\\right)=\\frac{1}{x+2}[\/latex] and [latex]\\displaystyle g\\left(x\\right)=\\frac{1}{x}-2[\/latex], are [latex]g[\/latex] and [latex]f[\/latex] inverses of each other?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q56557\">Show Answer<\/span><\/p>\n<div id=\"q56557\" class=\"hidden-answer\" style=\"display: none\">\n<p>Substitute [latex]\\displaystyle g(x)=\\frac{1}{x}-2[\/latex] into [latex]f(x)[\/latex], this means the new variable in\u00a0[latex]f(x)[\/latex] is [latex]\\displaystyle \\frac{1}{x}-2[\/latex] so you will substitute that expression where you see [latex]x[\/latex]. \u00a0Using parentheses helps keep track of things.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{cc} f\\left(\\displaystyle \\frac{1}{x}-2\\right)&=&\\displaystyle \\frac{1}{\\left(\\frac{1}{x}-2\\right)+2}\\\\&=&\\displaystyle \\frac{1}{\\frac{1}{x}}\\\\&=&{ x }\\hfill \\end{array}[\/latex]<\/p>\n<p>Next, we substitute [latex]\\displaystyle f(x)=\\frac{1}{x+2}[\/latex] into [latex]g(x)[\/latex], this means the new variable in\u00a0[latex]g(x)[\/latex] is [latex]\\displaystyle \\frac{1}{x+2}[\/latex] so you will substitute that expression where you see [latex]x[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{cc} g \\left(\\displaystyle \\frac{1}{x+2}\\right) &=&\\displaystyle \\frac{1}{\\left(\\frac{1}{x+2}\\right)}-2 \\\\ &=& x+2 -2 \\\\ &=& x \\hfill\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Yes, [latex]f[\/latex] and [latex]g[\/latex] are inverse functions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We will show one more example of how to use algebra to determine whether two functions are inverses of each other.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex 2: Determine if Two Functions Are Inverses\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/hzehBtNmw08?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/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-5115\">\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>Ex 1: Composition of Function. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/r_LssVS4NHk\">https:\/\/youtu.be\/r_LssVS4NHk<\/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>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/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":60342,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"},{\"type\":\"original\",\"description\":\"Ex 1: Composition of Function\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/r_LssVS4NHk\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen 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