{"id":15976,"date":"2019-12-05T05:17:11","date_gmt":"2019-12-05T05:17:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/chapter\/introduction-to-exponential-functions\/"},"modified":"2019-12-05T05:17:31","modified_gmt":"2019-12-05T05:17:31","slug":"introduction-to-exponential-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/chapter\/introduction-to-exponential-functions\/","title":{"raw":"Composite and Inverse Functions","rendered":"Composite and Inverse Functions"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Find composite functions<\/li>\n \t<li>Use compositions of functions to verify inverses algebraically<\/li>\n \t<li>Identify the domain and range of inverse functions with tables<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165134094620\">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<img class=\"aligncenter\" 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\">\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&nbsp;[latex]5[\/latex]th 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].&nbsp;By combining these two relationships into one function, we have performed function composition.<\/p>\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.\n\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.\n\n&nbsp;\n\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\">\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nUsing the functions provided, find [latex]f\\left(g\\left(x\\right)\\right)[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)[\/latex].\n\n[latex]f\\left(x\\right)=2x+1[\/latex]\n\n[latex]g\\left(x\\right)=3-x[\/latex]\n[reveal-answer q=\"337338\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"337338\"]\n\n[latex]f\\left(x\\right)=2x+1[\/latex]\n\n[latex]g\\left(x\\right)=3-x[\/latex]\n\nLet us begin by substituting [latex]g\\left(x\\right)[\/latex] into [latex]f\\left(x\\right)[\/latex].\n\n[latex]\\begin{array}{ll}f\\left(g\\left(x\\right)\\right) &amp; =2\\left(3-x\\right)+1\\hfill \\\\ &amp; =6 - 2x+1\\hfill \\\\ &amp; =7 - 2x\\hfill \\end{array}[\/latex]\n\nNow we can substitute [latex]f\\left(x\\right)[\/latex] into [latex]g\\left(x\\right)[\/latex].\n\n[latex]\\begin{array}{ll}g\\left(f\\left(x\\right)\\right) &amp; =3-\\left(2x+1\\right)\\hfill \\\\ &amp; =3 - 2x - 1\\hfill \\\\ &amp; =-2x+2\\hfill \\end{array}[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\nIn the following video, you will see another example of how to find the composition of two functions.\n\nhttps:\/\/youtu.be\/r_LssVS4NHk\n<h2>Inverse Functions<\/h2>\n<p id=\"fs-id1165137827441\">An <strong>inverse function<\/strong>&nbsp;is a function where 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]\\frac{1}{f\\left(x\\right)}[\/latex] because [latex]\\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\n<div id=\"fs-id1165137933105\" class=\"note textbox\">\n<h3 class=\"title\">Inverse Function<\/h3>\n<p id=\"fs-id1165137473076\">For any <strong>one-to-one function<\/strong> [latex]f\\left(x\\right)=y[\/latex], a function [latex]{f}^{-1}\\left(x\\right)[\/latex] is an <strong>inverse function<\/strong> of [latex]f[\/latex] if [latex]{f}^{-1}\\left(y\\right)=x[\/latex].<\/p>\n<p id=\"fs-id1165137444821\">The notation [latex]{f}^{-1}[\/latex] is read \"[latex]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]\nKeep in mind that<\/p>\n\n<div id=\"fs-id1165137581324\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]{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\n<\/div>\nIn our first example, we will identify an inverse function from ordered pairs.\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nIf 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?\n[reveal-answer q=\"664782\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"664782\"]\n<p id=\"fs-id1165137737081\">The inverse function reverses the input and output quantities, so if<\/p>\n\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&nbsp;\n\n[\/hidden-answer]\n\n<\/div>\n<h3>Analysis of the Solution<\/h3>\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 from the previous example in table form, the input and output are clearly reversed.<\/p>\n\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>[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<\/div>\nIn the following video, we show an example of finding corresponding input and output values given two ordered pairs from functions that are inverses.\n\nhttps:\/\/youtu.be\/IR_1L1mnpvw\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 \t<li>Substitute&nbsp;[latex]g(x)[\/latex]&nbsp;into [latex]f(x)[\/latex]. The result must be x. [latex]f\\left(g(x)\\right)=x[\/latex]<\/li>\n \t<li>Substitute&nbsp;[latex]f(x)[\/latex]&nbsp;into [latex]g(x)[\/latex]. The result must be x. [latex]g\\left(f(x)\\right)=x[\/latex]<\/li>\n<\/ol>\n<p style=\"text-align: center;\">If&nbsp;[latex]f(x)[\/latex] and&nbsp;&nbsp;[latex]g(x)[\/latex] are inverses, then&nbsp;&nbsp;[latex]f(x)=g^{-1}(x)[\/latex] and&nbsp;[latex]g(x)=f^{-1}(x)[\/latex]<\/p>\n\n<\/div>\nIn our next example, we will test inverse relationships algebraically.\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\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], is [latex]g[\/latex] the inverse of [latex]f[\/latex]?&nbsp; In other words, does [latex]g={f}^{-1}?[\/latex]\n[reveal-answer q=\"598434\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"598434\"]\n\nSubstitute [latex]g(x)=\\sqrt{x+3}[\/latex] into [latex]f(x)[\/latex]; this means the new variable in&nbsp;[latex]f(x)[\/latex] is [latex]\\sqrt{x+3}[\/latex], so you will substitute that expression where you see [latex]x[\/latex]. Using parentheses helps keep track of things.\n\n[latex]\\begin{array}{ll}f\\left(\\sqrt{x+3}\\right) &amp; ={(\\sqrt{x+3})}^2-3\\hfill\\\\ &amp; =x+3-3\\\\ &amp; =x\\hfill \\end{array}[\/latex]\n\nOur result implies that [latex]g(x)[\/latex] is indeed the inverse of&nbsp;[latex]f(x)[\/latex].\n\n[latex]g={f}^{-1}[\/latex],&nbsp;for [latex]x\\ge0[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\nIn the following video, we use algebra to determine if two functions are inverses.\n\nhttps:\/\/youtu.be\/vObCvTOatfQ\n\nWe will show one more example of how to verify whether you have an inverse algebraically.\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nIf [latex]f\\left(x\\right)=\\frac{1}{x+2}[\/latex] and [latex]g\\left(x\\right)=\\frac{1}{x}-2[\/latex], is [latex]g[\/latex] the inverse of [latex]f[\/latex]? In other words, does [latex]g={f}^{-1}?[\/latex]\n[reveal-answer q=\"56557\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"56557\"]\n\nSubstitute [latex]g(x)=\\frac{1}{x}-2[\/latex] into [latex]f(x)[\/latex]; this means the new variable in&nbsp;[latex]f(x)[\/latex] is [latex]\\frac{1}{x}-2[\/latex], so you will substitute that expression where you see [latex]x[\/latex]. Using parentheses helps keep track of things.\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} f\\left(\\frac{1}{x}-2\\right) &amp; =\\frac{1}{\\left(\\frac{1}{x}-2\\right)+2}\\hfill\\\\ &amp; =\\frac{1}{\\frac{1}{x}}\\hfill\\\\ &amp; ={ x }\\hfill \\end{array}[\/latex]<\/p>\n[latex]g={f}^{-1}[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\nWe will show one more example of how to use algebra to determine whether two functions are inverses of each other.\n\nhttps:\/\/youtu.be\/hzehBtNmw08\n<h2>&nbsp;Domain and Range of a Function and Its Inverse<\/h2>\nThe outputs of the function [latex]f[\/latex] are the inputs to [latex]{f}^{-1}[\/latex], so the range of [latex]f[\/latex] is also the domain of [latex]{f}^{-1}[\/latex]. Likewise, because the inputs to [latex]f[\/latex] are the outputs of [latex]{f}^{-1}[\/latex], the domain of [latex]f[\/latex] is the range of [latex]{f}^{-1}[\/latex]. We can visualize the situation.\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25200955\/CNX_Precalc_Figure_01_07_0032.jpg\" alt=\"Domain and range of a function and its inverse.\" width=\"487\" height=\"143\">\n\nIn many cases, if a function is not one-to-one, we can still restrict the function to a part of its domain on which it is one-to-one. For example, we can make a restricted version of the square function [latex]f\\left(x\\right)={x}^{2}[\/latex] with its domain limited to [latex]\\left[0,\\infty \\right)[\/latex], which is a one-to-one function (it passes the horizontal line test) and which has an inverse (the square-root function).\n<div class=\"textbox\">\n<h3 class=\"title\">Domain and Range of Inverse Functions<\/h3>\n<p id=\"fs-id1165135319550\">The range of a function [latex]f\\left(x\\right)[\/latex] is the domain of the inverse function [latex]{f}^{-1}\\left(x\\right)[\/latex].<\/p>\n<p id=\"fs-id1165137673886\">The domain of [latex]f\\left(x\\right)[\/latex] is the range of [latex]{f}^{-1}\\left(x\\right)[\/latex].<\/p>\n\n<\/div>\nIn our last example, we will define the domain and range of a function's inverse using a table of values, and evaluate the inverse at a specific value.\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p id=\"fs-id1165135435474\">A function [latex]f\\left(t\\right)[\/latex] is given&nbsp;below, showing distance in miles that a car has traveled in [latex]t[\/latex] minutes.<\/p>\n\n<ol>\n \t<li>Define the domain and range of the function and its inverse.<\/li>\n \t<li>Find and interpret [latex]{f}^{-1}\\left(70\\right)[\/latex].<\/li>\n<\/ol>\n<table style=\"width: 30%;\" summary=\"Two rows and five columns. The first row is labeled\">\n<tbody>\n<tr>\n<td><strong>[latex]t\\text{ (minutes)}[\/latex]<\/strong><\/td>\n<td>[latex]30[\/latex]<\/td>\n<td>[latex]50[\/latex]<\/td>\n<td>[latex]70[\/latex]<\/td>\n<td>[latex]90[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(t\\right)\\text{ (miles)}[\/latex] <\/strong><\/td>\n<td>[latex]20[\/latex]<\/td>\n<td>[latex]40[\/latex]<\/td>\n<td>[latex]60[\/latex]<\/td>\n<td>[latex]70[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n[reveal-answer q=\"713219\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"713219\"]\n\n1.<span style=\"text-decoration: underline;\">Domain and Range of the Original Function<\/span>\n\nThe domain of this tabular function, [latex]f\\left(t\\right)[\/latex], is all the input values, [latex]t[\/latex], in minutes:[latex]{30, 50, 70, 90}[\/latex]\n\nThe range of this tabular function, [latex]f\\left(t\\right)[\/latex], is all the output values [latex]f\\left(t\\right)[\/latex], in miles:[latex] {20, 40, 60, 70}[\/latex]\n\n<span style=\"text-decoration: underline;\">Domain and Range of the Inverse Function<\/span>\n<p id=\"fs-id1165137640334\">The domain for the inverse will be the outputs from the original, so the domain of &nbsp;[latex]{f}^{-1}(x)[\/latex] is the output values from&nbsp;[latex]f\\left(t\\right)[\/latex]:&nbsp;[latex]{20, 40, 60, 70}[\/latex]<\/p>\nThe range for the inverse will be the inputs from the original:&nbsp;[latex]{30, 50, 70, 90}[\/latex]\n\nThis translates to putting in a number of miles and getting out how long it took to drive that far in minutes.\n\n2.&nbsp;So in the expression [latex]{f}^{-1}\\left(70\\right)[\/latex],&nbsp;[latex]70[\/latex] is an output value of the original function, representing&nbsp;[latex]70[\/latex] miles. The inverse will return the corresponding input of the original function [latex]f[\/latex],&nbsp;[latex]90[\/latex] minutes, so [latex]{f}^{-1}\\left(70\\right)=90[\/latex]. The interpretation of this is that, to drive&nbsp;[latex]70[\/latex] miles, it took&nbsp;[latex]90[\/latex] minutes.[\/hidden-answer]\n\n<\/div>\n<h2>Summary<\/h2>\nThe inverse of a function can be defined for one-to-one functions. If a function is not one-to-one, it can be possible to restrict its domain to make it so. The domain of a function will become the range of its inverse. The range of a function will become the domain of its inverse. Inverses can be verified using tabular data as well as algebraically.\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find composite functions<\/li>\n<li>Use compositions of functions to verify inverses algebraically<\/li>\n<li>Identify the domain and range of inverse functions with tables<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165134094620\">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<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\/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 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&nbsp;[latex]5[\/latex]th 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].&nbsp;By 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 Solution<\/span><\/p>\n<div id=\"q337338\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f\\left(x\\right)=2x+1[\/latex]<\/p>\n<p>[latex]g\\left(x\\right)=3-x[\/latex]<\/p>\n<p>Let us begin by substituting [latex]g\\left(x\\right)[\/latex] into [latex]f\\left(x\\right)[\/latex].<\/p>\n<p>[latex]\\begin{array}{ll}f\\left(g\\left(x\\right)\\right) & =2\\left(3-x\\right)+1\\hfill \\\\ & =6 - 2x+1\\hfill \\\\ & =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>[latex]\\begin{array}{ll}g\\left(f\\left(x\\right)\\right) & =3-\\left(2x+1\\right)\\hfill \\\\ & =3 - 2x - 1\\hfill \\\\ & =-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<h2>Inverse Functions<\/h2>\n<p id=\"fs-id1165137827441\">An <strong>inverse function<\/strong>&nbsp;is a function where 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 &#8220;[latex]f[\/latex] inverse of [latex]x.[\/latex]&#8221; 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]\\frac{1}{f\\left(x\\right)}[\/latex] because [latex]\\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 <strong>one-to-one function<\/strong> [latex]f\\left(x\\right)=y[\/latex], a function [latex]{f}^{-1}\\left(x\\right)[\/latex] is an <strong>inverse function<\/strong> of [latex]f[\/latex] if [latex]{f}^{-1}\\left(y\\right)=x[\/latex].<\/p>\n<p id=\"fs-id1165137444821\">The notation [latex]{f}^{-1}[\/latex] is read &#8220;[latex]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 &#8220;[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]{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 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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q664782\">Show Solution<\/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<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Analysis of the Solution<\/h3>\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 from the previous example in 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>[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<\/div>\n<p>In the following video, we show an example of finding corresponding input and output values given two ordered pairs from functions that are inverses.<\/p>\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&nbsp;[latex]g(x)[\/latex]&nbsp;into [latex]f(x)[\/latex]. The result must be x. [latex]f\\left(g(x)\\right)=x[\/latex]<\/li>\n<li>Substitute&nbsp;[latex]f(x)[\/latex]&nbsp;into [latex]g(x)[\/latex]. The result must be x. [latex]g\\left(f(x)\\right)=x[\/latex]<\/li>\n<\/ol>\n<p style=\"text-align: center;\">If&nbsp;[latex]f(x)[\/latex] and&nbsp;&nbsp;[latex]g(x)[\/latex] are inverses, then&nbsp;&nbsp;[latex]f(x)=g^{-1}(x)[\/latex] and&nbsp;[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], is [latex]g[\/latex] the inverse of [latex]f[\/latex]?&nbsp; In other words, does [latex]g={f}^{-1}?[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q598434\">Show Solution<\/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&nbsp;[latex]f(x)[\/latex] is [latex]\\sqrt{x+3}[\/latex], so you will substitute that expression where you see [latex]x[\/latex]. Using parentheses helps keep track of things.<\/p>\n<p>[latex]\\begin{array}{ll}f\\left(\\sqrt{x+3}\\right) & ={(\\sqrt{x+3})}^2-3\\hfill\\\\ & =x+3-3\\\\ & =x\\hfill \\end{array}[\/latex]<\/p>\n<p>Our result implies that [latex]g(x)[\/latex] is indeed the inverse of&nbsp;[latex]f(x)[\/latex].<\/p>\n<p>[latex]g={f}^{-1}[\/latex],&nbsp;for [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 one more 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]f\\left(x\\right)=\\frac{1}{x+2}[\/latex] and [latex]g\\left(x\\right)=\\frac{1}{x}-2[\/latex], is [latex]g[\/latex] the inverse of [latex]f[\/latex]? In other words, does [latex]g={f}^{-1}?[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q56557\">Show Solution<\/span><\/p>\n<div id=\"q56557\" class=\"hidden-answer\" style=\"display: none\">\n<p>Substitute [latex]g(x)=\\frac{1}{x}-2[\/latex] into [latex]f(x)[\/latex]; this means the new variable in&nbsp;[latex]f(x)[\/latex] is [latex]\\frac{1}{x}-2[\/latex], so you will substitute that expression where you see [latex]x[\/latex]. Using parentheses helps keep track of things.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} f\\left(\\frac{1}{x}-2\\right) & =\\frac{1}{\\left(\\frac{1}{x}-2\\right)+2}\\hfill\\\\ & =\\frac{1}{\\frac{1}{x}}\\hfill\\\\ & ={ x }\\hfill \\end{array}[\/latex]<\/p>\n<p>[latex]g={f}^{-1}[\/latex]<\/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<h2>&nbsp;Domain and Range of a Function and Its Inverse<\/h2>\n<p>The outputs of the function [latex]f[\/latex] are the inputs to [latex]{f}^{-1}[\/latex], so the range of [latex]f[\/latex] is also the domain of [latex]{f}^{-1}[\/latex]. Likewise, because the inputs to [latex]f[\/latex] are the outputs of [latex]{f}^{-1}[\/latex], the domain of [latex]f[\/latex] is the range of [latex]{f}^{-1}[\/latex]. We can visualize the situation.<\/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\/25200955\/CNX_Precalc_Figure_01_07_0032.jpg\" alt=\"Domain and range of a function and its inverse.\" width=\"487\" height=\"143\" \/><\/p>\n<p>In many cases, if a function is not one-to-one, we can still restrict the function to a part of its domain on which it is one-to-one. For example, we can make a restricted version of the square function [latex]f\\left(x\\right)={x}^{2}[\/latex] with its domain limited to [latex]\\left[0,\\infty \\right)[\/latex], which is a one-to-one function (it passes the horizontal line test) and which has an inverse (the square-root function).<\/p>\n<div class=\"textbox\">\n<h3 class=\"title\">Domain and Range of Inverse Functions<\/h3>\n<p id=\"fs-id1165135319550\">The range of a function [latex]f\\left(x\\right)[\/latex] is the domain of the inverse function [latex]{f}^{-1}\\left(x\\right)[\/latex].<\/p>\n<p id=\"fs-id1165137673886\">The domain of [latex]f\\left(x\\right)[\/latex] is the range of [latex]{f}^{-1}\\left(x\\right)[\/latex].<\/p>\n<\/div>\n<p>In our last example, we will define the domain and range of a function&#8217;s inverse using a table of values, and evaluate the inverse at a specific value.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p id=\"fs-id1165135435474\">A function [latex]f\\left(t\\right)[\/latex] is given&nbsp;below, showing distance in miles that a car has traveled in [latex]t[\/latex] minutes.<\/p>\n<ol>\n<li>Define the domain and range of the function and its inverse.<\/li>\n<li>Find and interpret [latex]{f}^{-1}\\left(70\\right)[\/latex].<\/li>\n<\/ol>\n<table style=\"width: 30%;\" summary=\"Two rows and five columns. The first row is labeled\">\n<tbody>\n<tr>\n<td><strong>[latex]t\\text{ (minutes)}[\/latex]<\/strong><\/td>\n<td>[latex]30[\/latex]<\/td>\n<td>[latex]50[\/latex]<\/td>\n<td>[latex]70[\/latex]<\/td>\n<td>[latex]90[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(t\\right)\\text{ (miles)}[\/latex] <\/strong><\/td>\n<td>[latex]20[\/latex]<\/td>\n<td>[latex]40[\/latex]<\/td>\n<td>[latex]60[\/latex]<\/td>\n<td>[latex]70[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q713219\">Show Solution<\/span><\/p>\n<div id=\"q713219\" class=\"hidden-answer\" style=\"display: none\">\n<p>1.<span style=\"text-decoration: underline;\">Domain and Range of the Original Function<\/span><\/p>\n<p>The domain of this tabular function, [latex]f\\left(t\\right)[\/latex], is all the input values, [latex]t[\/latex], in minutes:[latex]{30, 50, 70, 90}[\/latex]<\/p>\n<p>The range of this tabular function, [latex]f\\left(t\\right)[\/latex], is all the output values [latex]f\\left(t\\right)[\/latex], in miles:[latex]{20, 40, 60, 70}[\/latex]<\/p>\n<p><span style=\"text-decoration: underline;\">Domain and Range of the Inverse Function<\/span><\/p>\n<p id=\"fs-id1165137640334\">The domain for the inverse will be the outputs from the original, so the domain of &nbsp;[latex]{f}^{-1}(x)[\/latex] is the output values from&nbsp;[latex]f\\left(t\\right)[\/latex]:&nbsp;[latex]{20, 40, 60, 70}[\/latex]<\/p>\n<p>The range for the inverse will be the inputs from the original:&nbsp;[latex]{30, 50, 70, 90}[\/latex]<\/p>\n<p>This translates to putting in a number of miles and getting out how long it took to drive that far in minutes.<\/p>\n<p>2.&nbsp;So in the expression [latex]{f}^{-1}\\left(70\\right)[\/latex],&nbsp;[latex]70[\/latex] is an output value of the original function, representing&nbsp;[latex]70[\/latex] miles. The inverse will return the corresponding input of the original function [latex]f[\/latex],&nbsp;[latex]90[\/latex] minutes, so [latex]{f}^{-1}\\left(70\\right)=90[\/latex]. The interpretation of this is that, to drive&nbsp;[latex]70[\/latex] miles, it took&nbsp;[latex]90[\/latex] minutes.<\/p><\/div>\n<\/div>\n<\/div>\n<h2>Summary<\/h2>\n<p>The inverse of a function can be defined for one-to-one functions. If a function is not one-to-one, it can be possible to restrict its domain to make it so. The domain of a function will become the range of its inverse. The range of a function will become the domain of its inverse. Inverses can be verified using tabular data as well as algebraically.<\/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-15976\">\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: Determine if Two Functions Are Inverses. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/vObCvTOatfQ\">https:\/\/youtu.be\/vObCvTOatfQ<\/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>Ex 2: Determine if Two Functions Are Inverses. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/hzehBtNmw08\">https:\/\/youtu.be\/hzehBtNmw08<\/a>. <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><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>Ex: Function and Inverse Function Values. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/IR_1L1mnpvw\">https:\/\/youtu.be\/IR_1L1mnpvw<\/a>. <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 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