{"id":220,"date":"2023-11-08T16:10:29","date_gmt":"2023-11-08T16:10:29","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/introduction-to-exponential-functions\/"},"modified":"2026-02-13T22:26:33","modified_gmt":"2026-02-13T22:26:33","slug":"7-1-composing-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-interalgebra\/chapter\/7-1-composing-functions\/","title":{"raw":"7.1 Composing Functions","rendered":"7.1 Composing Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Evaluate composite functions.<\/li>\r\n \t<li>Find composite functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\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>\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\/25200743\/CNX_Precalc_Figure_01_04_0062.jpg\" alt=\"Expression: C(T(5)). A bracket above the full expression is labeled 'Cost for the temperature'. An arrow points below the expression to T of 5 and is labeled 'Temperature on day 5'.\" width=\"487\" height=\"140\" \/>\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\u00a0[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].\u00a0By combining these two relationships into one function, we have performed function composition.<\/p>\r\n\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Composition of Functions<\/h3>\r\nThe <em>composite function<\/em> [latex]f \\circ g[\/latex], the <em>composition<\/em> of [latex]f[\/latex] and [latex]g[\/latex], is defined as [latex](f \\circ g)(x) \\ =\\ f(g(x))[\/latex]\r\n\r\n<\/div>\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 open circle symbol [latex]\\circ [\/latex] is called the composition operator.\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=\"the composition f composed with g at x is f of g of x. g(x), the output of g is the input of f. x is the input of g.\" width=\"487\" height=\"171\" \/>\r\nIt is important to correctly apply the order of operations when 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\nOnce we compose a new function from two existing functions, we need to be able to evaluate it for any input in its domain. We will do this with specific numerical inputs for functions expressed as tables, graphs, and formulas and with variables as inputs to functions expressed as formulas. In each case we evaluate the inner function using the starting input and then use the inner function\u2019s output as the input for the outer function.\r\n<h2>Evaluating Composite Functions Using Formulas<\/h2>\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nGiven [latex]f\\left(x\\right)=-3x-2[\/latex] and [latex]g\\left(x\\right)=x+5[\/latex], evaluate [latex]\\left( f \\circ g \\right) [\/latex] for [latex] x=2[\/latex].\r\n\r\n[latex]\\left( f \\circ g \\right)\\left(2\\right) = f\\left(g\\left(2\\right)\\right)[\/latex].\r\n\r\n[reveal-answer q=\"337568\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"337568\"]\r\n\r\nTo find [latex]f\\left(g\\left(2\\right)\\right)[\/latex], first substitute [latex]\\require{color}{\\color{Green}{2}}[\/latex] into [latex]g\\left(x\\right)[\/latex].\r\n\r\n[latex]\r\n\\begin{align}\r\n\r\ng\\left( {\\color{Green}{2}}\\right) &amp; =\\left( {\\color{Green}{2}} \\right)+5 &amp;&amp; \\color{blue}{\\textsf{substitute $x=2$}}\\\\[5pt]\r\n\r\n&amp;= 2+5\\\\[5pt]\r\n\r\n&amp;=7 \\\\[5pt]\r\n\r\n\\end{align}[\/latex]\r\n\r\nNext, substitute [latex]{\\color{Green}{g\\left(2\\right)}} [\/latex] into [latex]f\\left(x\\right)[\/latex].\r\n\r\n[latex]\r\n\\begin{align}\r\n\r\nf\\left( {\\color{Green} {g\\left(2\\right) }} \\right) &amp; = f\\left( {\\color{Green} {7} }\\right) &amp;&amp; \\color{blue}{\\textsf{substitute $g\\left(2\\right)=7$}} \\\\[5pt]\r\n\r\n&amp;=-3\\left({\\color{Green}{7}}\\right)-2\\\\[5pt]\r\n\r\n&amp;= -21-2\\\\[5pt]\r\n\r\n&amp;= -23 \\\\[5pt]\r\n\r\n\\end{align}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nGiven [latex]f\\left(x\\right)=2x+1[\/latex] and [latex]g\\left(x\\right)=3-x[\/latex], find [latex]f\\left(g\\left(x\\right)\\right)[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)[\/latex].\r\n\r\n[reveal-answer q=\"337338\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"337338\"]\r\n\r\nTo find [latex]f\\left(g\\left(x\\right)\\right)[\/latex], substitute [latex]\\require{color}{\\color{Green}{g\\left(x\\right)}}[\/latex] into [latex]f\\left(x\\right)[\/latex].\r\n\r\n[latex]\r\n\\begin{align}\r\n\r\nf\\left( {\\color{Green}{g\\left(x\\right)}}\\right) \\\\[5pt]&amp; =2\\left({\\color{Green}{3-x}} \\right)+1 &amp;&amp; \\color{blue}{\\textsf{substitute $g\\left(x\\right)=3-x$}}\\\\[5pt]\r\n\r\n&amp;= 6 - 2x+1\\\\[5pt]\r\n\r\n&amp;=7 - 2x\\\\[5pt]\r\n\r\n\\end{align}[\/latex]\r\n\r\nTo find [latex]g\\left(f\\left(x\\right)\\right)[\/latex], substitute [latex]{\\color{Green}{f\\left(x\\right)}} [\/latex] into [latex]g\\left(x\\right)[\/latex].\r\n\r\n[latex]\r\n\\begin{align}\r\n\r\ng\\left({\\color{Green}{f\\left(x\\right)}}\\right) &amp; =3-\\left({\\color{Green}{2x+1}}\\right) &amp;&amp; \\color{blue}{\\textsf{substitute $f\\left(x\\right)=2x+1$}}\\\\[5pt]\r\n\r\n&amp;= 3 - 2x - 1\\\\[5pt]\r\n\r\n&amp;= -2x+2 \\\\[5pt]\r\n\r\n\\end{align}[\/latex]\r\n\r\nThis example demonstrates that, in general, [latex]\\left(f\\circ g\\right)\\neq\\left( g \\circ f\\right) [\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\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\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>EXAMPLE<\/h3>\r\nGiven [latex]f\\left(x\\right)=4x+3[\/latex] and [latex]g\\left(x\\right)=4x^2+3x-1[\/latex], find [latex]\\left(f\\circ g\\right)[\/latex] and [latex]\\left(g\\circ f\\right)[\/latex].\r\n\r\n[reveal-answer q=\"347338\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"347338\"]\r\n\r\n[latex]\r\n\\begin{align}\r\n\r\n\\left( f \\circ g \\right)\\left(x\\right) = f\\left({\\color{Green}{g\\left(x\\right)}}\\right)\\\\[5pt]\r\n\r\n&amp;= f\\left({\\color{Green}{4x^2+3x-1}}\\right) &amp;&amp; \\color{blue}{\\textsf{substitute $g\\left(x\\right)=4x^2+3x-1$}}\\\\[5pt]\r\n\r\n&amp;=4\\left({\\color{Green}{4x^2+3x-1}}\\right)+3\\\\[5pt]\r\n\r\n&amp;=16x^2+12x-4+3\\\\[5pt]\r\n\r\n&amp;=16x^2+12x-1\\\\[5pt]\r\n\r\n\\end{align}[\/latex]\r\n\r\n&nbsp;\r\n\r\n[latex]\r\n\\begin{align}\r\n\r\n\\left( g \\circ f \\right)\\left(x\\right) = g\\left({\\color{Green}{f\\left(x\\right)}}\\right)\\\\[5pt]\r\n\r\n&amp;= g\\left({\\color{Green}{4x+3}}\\right) &amp;&amp; \\color{blue}{\\textsf{substitute $f\\left(x\\right)=4x+3$}}\\\\[5pt]\r\n\r\n&amp;=4\\left({\\color{Green}{4x+3}}\\right)^2+3\\left({\\color{Green}{4x+3}}\\right)-1\\\\[5pt]\r\n\r\n&amp;=4\\left(4x+3\\right)\\left(4x+3\\right)+12x + 9-1\\\\[5pt]\r\n\r\n&amp;=4\\left(16x^2+12x+12x+9\\right)+12x + 8\\\\[5pt]\r\n\r\n&amp;=4\\left(16x^2+24x+9\\right)+12x + 8\\\\[5pt]\r\n\r\n&amp;=64x^2+96x+36+12x + 8\\\\[5pt]\r\n\r\n&amp;=64x^2+108x+44\\\\[5pt]\r\n\r\n\\end{align}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Evaluating Composite Functions Using Tables<\/h2>\r\nWhen working with functions given as tables, we read input and output values from the table entries and always work from the inside to the outside. We evaluate the inside function first and then use the output of the inside function as the input to the outside function.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using a Table to Evaluate a Composite Function<\/h3>\r\nUsing the table below,\u00a0evaluate [latex]f\\left(g\\left(3\\right)\\right)[\/latex] and [latex]g\\left(f\\left(3\\right)\\right)[\/latex].\r\n<table style=\"width: 371px;\" summary=\"Five rows and three columns. The first column is labeled,\"><colgroup> <col \/> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"width: 112.65px; text-align: center;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 176.35px; text-align: center;\">[latex]f\\left(x\\right)[\/latex]<\/th>\r\n<th style=\"width: 182.6px; text-align: center;\">[latex]g\\left(x\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 112.65px; text-align: center;\">1<\/td>\r\n<td style=\"width: 176.35px; text-align: center;\">6<\/td>\r\n<td style=\"width: 182.6px; text-align: center;\">3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 112.65px; text-align: center;\">2<\/td>\r\n<td style=\"width: 176.35px; text-align: center;\">8<\/td>\r\n<td style=\"width: 182.6px; text-align: center;\">5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 112.65px; text-align: center;\">3<\/td>\r\n<td style=\"width: 176.35px; text-align: center;\">3<\/td>\r\n<td style=\"width: 182.6px; text-align: center;\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 112.65px; text-align: center;\">4<\/td>\r\n<td style=\"width: 176.35px; text-align: center;\">1<\/td>\r\n<td style=\"width: 182.6px; text-align: center;\">7<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"195147\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"195147\"]\r\n\r\nTo evaluate [latex]f\\left(g\\left(3\\right)\\right)[\/latex], start from the inside with the input value [latex]x=3[\/latex]. Then evaluate the \"inside\" expression [latex]g\\left(3\\right)[\/latex] using the column in the table that defines the function [latex]g:[\/latex] [latex]g\\left(3\\right)=2[\/latex]. Use that result as the input to the function [latex]f[\/latex]. Thus [latex]g\\left(3\\right)=2[\/latex] and [latex]f\\left(g\\left(3\\right)\\right)=f\\left(2\\right)[\/latex]. Next, using the column in the table that defines the function [latex]f[\/latex], we find that [latex]f\\left(2\\right)=8[\/latex].\r\n<p style=\"text-align: center;\">[latex]f\\left(g\\left(3\\right)\\right)=f\\left(2\\right)=8[\/latex]<\/p>\r\nTo evaluate [latex]g\\left(f\\left(3\\right)\\right)[\/latex], first evaluate the \"inside\" expression [latex]f\\left(3\\right)[\/latex] using the column in the table that defines the function [latex]f[\/latex]: [latex]f\\left(3\\right)=3[\/latex]. Next, use the column in the table for [latex]g[\/latex] to evaluate [latex]g\\left(3\\right)=2[\/latex].\r\n<p style=\"text-align: center;\">[latex]g\\left(f\\left(3\\right)\\right)=g\\left(3\\right)=2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUsing the table below, evaluate [latex]f\\left(g\\left(1\\right)\\right)[\/latex] and [latex]g\\left(f\\left(4\\right)\\right)[\/latex].\r\n<table style=\"height: 60px; width: 328px;\" summary=\"Five rows and three columns. The first column is labeled,\"><colgroup> <col \/> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"text-align: center; height: 12px; width: 128.783px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"text-align: center; height: 12px; width: 188.883px;\">[latex]f\\left(x\\right)[\/latex]<\/th>\r\n<th style=\"text-align: center; height: 12px; width: 153.933px;\">[latex]g\\left(x\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 128.783px;\">1<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 188.883px;\">6<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 153.933px;\">3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 128.783px;\">2<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 188.883px;\">8<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 153.933px;\">5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 128.783px;\">3<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 188.883px;\">3<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 153.933px;\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 128.783px;\">4<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 188.883px;\">1<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 153.933px;\">7<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"161706\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"161706\"]\r\n\r\n[latex]f\\left(g\\left(1\\right)\\right)=f\\left(3\\right)=3[\/latex] and [latex]g\\left(f\\left(4\\right)\\right)=g\\left(1\\right)=3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nPractice using the table to evaluate the functions below.\r\n\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=3585&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Evaluating Composite Functions Using Graphs<\/h2>\r\nGiven individual functions as graphs, evaluation of composite functions is similar to evaluation using tables, but the input and output values are found from the [latex]x\\text{-}[\/latex] and [latex]y\\text{-}[\/latex] axes of the graphs.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using a Graph to Evaluate a Composite Function<\/h3>\r\nUsing the graphs below, evaluate [latex]f\\left(g\\left(1\\right)\\right)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18195618\/CNX_Precalc_Figure_01_04_002ab2.jpg\" alt=\"Two graphs labeled g of x and f of x on different coordinate planes. G of x is a right side up parabola with vertex (3, negative 1) and additional points (1,3), (2,0), (4,0) and (5,3). F of x is an upside down parabola with vertex (3,6) and additional points (1,2) (5,2), (0, negative 3), and (6, negative 3).\" width=\"975\" height=\"543\" \/>\r\n\r\n[reveal-answer q=\"44734\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44734\"]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18195620\/CNX_Precalc_Figure_01_04_0042.jpg\" alt=\"Two graphs labeled g of x and f of x on different coordinate planes. (g of x) is a positive parabola and (f of x) is a negative parabola. Points are plotted: (1,3) on g of x and (3,6) on f of x. Equations g of 1 = 3 and f of 3 = 6 are written respectively below each graph.\" width=\"975\" height=\"543\" \/>\r\n\r\nTo evaluate [latex]f\\left(g\\left(1\\right)\\right)[\/latex], start with the \"inside\" evaluation.<span id=\"fs-id1165137644158\">\r\n<\/span>\r\n\r\nEvaluate [latex]g\\left(1\\right)[\/latex] using the graph of [latex]g\\left(x\\right)[\/latex] by finding the input of 1 on the [latex]x\\text{-}[\/latex] axis and finding the output value of the graph at that input. Here, [latex]g\\left(1\\right)=3[\/latex]. Use this value as the input to the function [latex]f[\/latex].\r\n<p style=\"text-align: center;\">[latex]f\\left(g\\left(1\\right)\\right)=f\\left(3\\right)[\/latex]<\/p>\r\nTo evaluate the composite function, look at the graph of [latex]f\\left(x\\right)[\/latex], finding the input of 3 on the [latex]x\\text{-}[\/latex] axis and reading the output value of the graph at this input. Here, [latex]f\\left(3\\right)=6[\/latex], so [latex]f\\left(g\\left(1\\right)\\right)=6[\/latex].\r\n\r\nThe figure\u00a0shows how we can mark the graphs with arrows to trace the path from the input value to the output value.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18195623\/CNX_Precalc_Figure_01_04_0052.jpg\" alt=\"Two graphs labeled g of x and f of x on different coordinate planes. (g of x) is a positive parabola and (f of x) is a negative parabola. On g of x an arrow points from 1 on x-axis up to (1,3) and over to 3 on y-axis. On f of x an arrow points from 3 on x-axis to (3,6) and over to 6 on y-axis. \" width=\"975\" height=\"520\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Evaluate composite functions.<\/li>\n<li>Find composite functions.<\/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=\"Expression: C(T(5)). A bracket above the full expression is labeled 'Cost for the temperature'. An arrow points below the expression to T of 5 and is labeled '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\u00a0[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].\u00a0By combining these two relationships into one function, we have performed function composition.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Composition of Functions<\/h3>\n<p>The <em>composite function<\/em> [latex]f \\circ g[\/latex], the <em>composition<\/em> of [latex]f[\/latex] and [latex]g[\/latex], is defined as [latex](f \\circ g)(x) \\ =\\ f(g(x))[\/latex]<\/p>\n<\/div>\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 open circle symbol [latex]\\circ[\/latex] is called the composition operator.<\/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=\"the composition f composed with g at x is f of g of x. g(x), the output of g is the input of f. x is the input of g.\" width=\"487\" height=\"171\" \/><br \/>\nIt is important to correctly apply the order of operations when 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>Once we compose a new function from two existing functions, we need to be able to evaluate it for any input in its domain. We will do this with specific numerical inputs for functions expressed as tables, graphs, and formulas and with variables as inputs to functions expressed as formulas. In each case we evaluate the inner function using the starting input and then use the inner function\u2019s output as the input for the outer function.<\/p>\n<h2>Evaluating Composite Functions Using Formulas<\/h2>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>Given [latex]f\\left(x\\right)=-3x-2[\/latex] and [latex]g\\left(x\\right)=x+5[\/latex], evaluate [latex]\\left( f \\circ g \\right)[\/latex] for [latex]x=2[\/latex].<\/p>\n<p>[latex]\\left( f \\circ g \\right)\\left(2\\right) = f\\left(g\\left(2\\right)\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q337568\">Show Solution<\/span><\/p>\n<div id=\"q337568\" class=\"hidden-answer\" style=\"display: none\">\n<p>To find [latex]f\\left(g\\left(2\\right)\\right)[\/latex], first substitute [latex]\\require{color}{\\color{Green}{2}}[\/latex] into [latex]g\\left(x\\right)[\/latex].<\/p>\n<p>[latex]\\begin{align}    g\\left( {\\color{Green}{2}}\\right) & =\\left( {\\color{Green}{2}} \\right)+5 && \\color{blue}{\\textsf{substitute $x=2$}}\\\\[5pt]    &= 2+5\\\\[5pt]    &=7 \\\\[5pt]    \\end{align}[\/latex]<\/p>\n<p>Next, substitute [latex]{\\color{Green}{g\\left(2\\right)}}[\/latex] into [latex]f\\left(x\\right)[\/latex].<\/p>\n<p>[latex]\\begin{align}    f\\left( {\\color{Green} {g\\left(2\\right) }} \\right) & = f\\left( {\\color{Green} {7} }\\right) && \\color{blue}{\\textsf{substitute $g\\left(2\\right)=7$}} \\\\[5pt]    &=-3\\left({\\color{Green}{7}}\\right)-2\\\\[5pt]    &= -21-2\\\\[5pt]    &= -23 \\\\[5pt]    \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Given [latex]f\\left(x\\right)=2x+1[\/latex] and [latex]g\\left(x\\right)=3-x[\/latex], find [latex]f\\left(g\\left(x\\right)\\right)[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)[\/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>To find [latex]f\\left(g\\left(x\\right)\\right)[\/latex], substitute [latex]\\require{color}{\\color{Green}{g\\left(x\\right)}}[\/latex] into [latex]f\\left(x\\right)[\/latex].<\/p>\n<p>[latex]\\begin{align}    f\\left( {\\color{Green}{g\\left(x\\right)}}\\right) \\\\[5pt]& =2\\left({\\color{Green}{3-x}} \\right)+1 && \\color{blue}{\\textsf{substitute $g\\left(x\\right)=3-x$}}\\\\[5pt]    &= 6 - 2x+1\\\\[5pt]    &=7 - 2x\\\\[5pt]    \\end{align}[\/latex]<\/p>\n<p>To find [latex]g\\left(f\\left(x\\right)\\right)[\/latex], substitute [latex]{\\color{Green}{f\\left(x\\right)}}[\/latex] into [latex]g\\left(x\\right)[\/latex].<\/p>\n<p>[latex]\\begin{align}    g\\left({\\color{Green}{f\\left(x\\right)}}\\right) & =3-\\left({\\color{Green}{2x+1}}\\right) && \\color{blue}{\\textsf{substitute $f\\left(x\\right)=2x+1$}}\\\\[5pt]    &= 3 - 2x - 1\\\\[5pt]    &= -2x+2 \\\\[5pt]    \\end{align}[\/latex]<\/p>\n<p>This example demonstrates that, in general, [latex]\\left(f\\circ g\\right)\\neq\\left( g \\circ f\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\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<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>EXAMPLE<\/h3>\n<p>Given [latex]f\\left(x\\right)=4x+3[\/latex] and [latex]g\\left(x\\right)=4x^2+3x-1[\/latex], find [latex]\\left(f\\circ g\\right)[\/latex] and [latex]\\left(g\\circ f\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q347338\">Show Solution<\/span><\/p>\n<div id=\"q347338\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{align}    \\left( f \\circ g \\right)\\left(x\\right) = f\\left({\\color{Green}{g\\left(x\\right)}}\\right)\\\\[5pt]    &= f\\left({\\color{Green}{4x^2+3x-1}}\\right) && \\color{blue}{\\textsf{substitute $g\\left(x\\right)=4x^2+3x-1$}}\\\\[5pt]    &=4\\left({\\color{Green}{4x^2+3x-1}}\\right)+3\\\\[5pt]    &=16x^2+12x-4+3\\\\[5pt]    &=16x^2+12x-1\\\\[5pt]    \\end{align}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>[latex]\\begin{align}    \\left( g \\circ f \\right)\\left(x\\right) = g\\left({\\color{Green}{f\\left(x\\right)}}\\right)\\\\[5pt]    &= g\\left({\\color{Green}{4x+3}}\\right) && \\color{blue}{\\textsf{substitute $f\\left(x\\right)=4x+3$}}\\\\[5pt]    &=4\\left({\\color{Green}{4x+3}}\\right)^2+3\\left({\\color{Green}{4x+3}}\\right)-1\\\\[5pt]    &=4\\left(4x+3\\right)\\left(4x+3\\right)+12x + 9-1\\\\[5pt]    &=4\\left(16x^2+12x+12x+9\\right)+12x + 8\\\\[5pt]    &=4\\left(16x^2+24x+9\\right)+12x + 8\\\\[5pt]    &=64x^2+96x+36+12x + 8\\\\[5pt]    &=64x^2+108x+44\\\\[5pt]    \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Evaluating Composite Functions Using Tables<\/h2>\n<p>When working with functions given as tables, we read input and output values from the table entries and always work from the inside to the outside. We evaluate the inside function first and then use the output of the inside function as the input to the outside function.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Using a Table to Evaluate a Composite Function<\/h3>\n<p>Using the table below,\u00a0evaluate [latex]f\\left(g\\left(3\\right)\\right)[\/latex] and [latex]g\\left(f\\left(3\\right)\\right)[\/latex].<\/p>\n<table style=\"width: 371px;\" summary=\"Five rows and three columns. The first column is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr>\n<th style=\"width: 112.65px; text-align: center;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 176.35px; text-align: center;\">[latex]f\\left(x\\right)[\/latex]<\/th>\n<th style=\"width: 182.6px; text-align: center;\">[latex]g\\left(x\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 112.65px; text-align: center;\">1<\/td>\n<td style=\"width: 176.35px; text-align: center;\">6<\/td>\n<td style=\"width: 182.6px; text-align: center;\">3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 112.65px; text-align: center;\">2<\/td>\n<td style=\"width: 176.35px; text-align: center;\">8<\/td>\n<td style=\"width: 182.6px; text-align: center;\">5<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 112.65px; text-align: center;\">3<\/td>\n<td style=\"width: 176.35px; text-align: center;\">3<\/td>\n<td style=\"width: 182.6px; text-align: center;\">2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 112.65px; text-align: center;\">4<\/td>\n<td style=\"width: 176.35px; text-align: center;\">1<\/td>\n<td style=\"width: 182.6px; text-align: center;\">7<\/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=\"q195147\">Show Solution<\/span><\/p>\n<div id=\"q195147\" class=\"hidden-answer\" style=\"display: none\">\n<p>To evaluate [latex]f\\left(g\\left(3\\right)\\right)[\/latex], start from the inside with the input value [latex]x=3[\/latex]. Then evaluate the &#8220;inside&#8221; expression [latex]g\\left(3\\right)[\/latex] using the column in the table that defines the function [latex]g:[\/latex] [latex]g\\left(3\\right)=2[\/latex]. Use that result as the input to the function [latex]f[\/latex]. Thus [latex]g\\left(3\\right)=2[\/latex] and [latex]f\\left(g\\left(3\\right)\\right)=f\\left(2\\right)[\/latex]. Next, using the column in the table that defines the function [latex]f[\/latex], we find that [latex]f\\left(2\\right)=8[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(g\\left(3\\right)\\right)=f\\left(2\\right)=8[\/latex]<\/p>\n<p>To evaluate [latex]g\\left(f\\left(3\\right)\\right)[\/latex], first evaluate the &#8220;inside&#8221; expression [latex]f\\left(3\\right)[\/latex] using the column in the table that defines the function [latex]f[\/latex]: [latex]f\\left(3\\right)=3[\/latex]. Next, use the column in the table for [latex]g[\/latex] to evaluate [latex]g\\left(3\\right)=2[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]g\\left(f\\left(3\\right)\\right)=g\\left(3\\right)=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Using the table below, evaluate [latex]f\\left(g\\left(1\\right)\\right)[\/latex] and [latex]g\\left(f\\left(4\\right)\\right)[\/latex].<\/p>\n<table style=\"height: 60px; width: 328px;\" summary=\"Five rows and three columns. The first column is labeled,\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr style=\"height: 12px;\">\n<th style=\"text-align: center; height: 12px; width: 128.783px;\">[latex]x[\/latex]<\/th>\n<th style=\"text-align: center; height: 12px; width: 188.883px;\">[latex]f\\left(x\\right)[\/latex]<\/th>\n<th style=\"text-align: center; height: 12px; width: 153.933px;\">[latex]g\\left(x\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 128.783px;\">1<\/td>\n<td style=\"text-align: center; height: 12px; width: 188.883px;\">6<\/td>\n<td style=\"text-align: center; height: 12px; width: 153.933px;\">3<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 128.783px;\">2<\/td>\n<td style=\"text-align: center; height: 12px; width: 188.883px;\">8<\/td>\n<td style=\"text-align: center; height: 12px; width: 153.933px;\">5<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 128.783px;\">3<\/td>\n<td style=\"text-align: center; height: 12px; width: 188.883px;\">3<\/td>\n<td style=\"text-align: center; height: 12px; width: 153.933px;\">2<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 128.783px;\">4<\/td>\n<td style=\"text-align: center; height: 12px; width: 188.883px;\">1<\/td>\n<td style=\"text-align: center; height: 12px; width: 153.933px;\">7<\/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=\"q161706\">Show Solution<\/span><\/p>\n<div id=\"q161706\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f\\left(g\\left(1\\right)\\right)=f\\left(3\\right)=3[\/latex] and [latex]g\\left(f\\left(4\\right)\\right)=g\\left(1\\right)=3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Practice using the table to evaluate the functions below.<\/p>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=3585&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"350\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<h2>Evaluating Composite Functions Using Graphs<\/h2>\n<p>Given individual functions as graphs, evaluation of composite functions is similar to evaluation using tables, but the input and output values are found from the [latex]x\\text{-}[\/latex] and [latex]y\\text{-}[\/latex] axes of the graphs.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Using a Graph to Evaluate a Composite Function<\/h3>\n<p>Using the graphs below, evaluate [latex]f\\left(g\\left(1\\right)\\right)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18195618\/CNX_Precalc_Figure_01_04_002ab2.jpg\" alt=\"Two graphs labeled g of x and f of x on different coordinate planes. G of x is a right side up parabola with vertex (3, negative 1) and additional points (1,3), (2,0), (4,0) and (5,3). F of x is an upside down parabola with vertex (3,6) and additional points (1,2) (5,2), (0, negative 3), and (6, negative 3).\" width=\"975\" height=\"543\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44734\">Show Solution<\/span><\/p>\n<div id=\"q44734\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18195620\/CNX_Precalc_Figure_01_04_0042.jpg\" alt=\"Two graphs labeled g of x and f of x on different coordinate planes. (g of x) is a positive parabola and (f of x) is a negative parabola. Points are plotted: (1,3) on g of x and (3,6) on f of x. Equations g of 1 = 3 and f of 3 = 6 are written respectively below each graph.\" width=\"975\" height=\"543\" \/><\/p>\n<p>To evaluate [latex]f\\left(g\\left(1\\right)\\right)[\/latex], start with the &#8220;inside&#8221; evaluation.<span id=\"fs-id1165137644158\"><br \/>\n<\/span><\/p>\n<p>Evaluate [latex]g\\left(1\\right)[\/latex] using the graph of [latex]g\\left(x\\right)[\/latex] by finding the input of 1 on the [latex]x\\text{-}[\/latex] axis and finding the output value of the graph at that input. Here, [latex]g\\left(1\\right)=3[\/latex]. Use this value as the input to the function [latex]f[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(g\\left(1\\right)\\right)=f\\left(3\\right)[\/latex]<\/p>\n<p>To evaluate the composite function, look at the graph of [latex]f\\left(x\\right)[\/latex], finding the input of 3 on the [latex]x\\text{-}[\/latex] axis and reading the output value of the graph at this input. Here, [latex]f\\left(3\\right)=6[\/latex], so [latex]f\\left(g\\left(1\\right)\\right)=6[\/latex].<\/p>\n<p>The figure\u00a0shows how we can mark the graphs with arrows to trace the path from the input value to the output value.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18195623\/CNX_Precalc_Figure_01_04_0052.jpg\" alt=\"Two graphs labeled g of x and f of x on different coordinate planes. (g of x) is a positive parabola and (f of x) is a negative parabola. On g of x an arrow points from 1 on x-axis up to (1,3) and over to 3 on y-axis. On f of x an arrow points from 3 on x-axis to (3,6) and over to 6 on y-axis.\" width=\"975\" height=\"520\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\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-220\">\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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