{"id":146,"date":"2023-06-21T13:22:37","date_gmt":"2023-06-21T13:22:37","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/evaluate-a-composition-of-functions\/"},"modified":"2023-10-19T22:33:59","modified_gmt":"2023-10-19T22:33:59","slug":"evaluate-a-composition-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/evaluate-a-composition-of-functions\/","title":{"raw":"\u25aa   Evaluate a Composition of Functions","rendered":"\u25aa   Evaluate a Composition of Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Evaluate a composition of functions using a table.<\/li>\r\n \t<li>Evaluate a composition of functions using an equation.<\/li>\r\n<\/ul>\r\n<\/div>\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 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: 30%;\" 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>[latex]x[\/latex]<\/th>\r\n<th>[latex]f\\left(x\\right)[\/latex]<\/th>\r\n<th>[latex]g\\left(x\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>1<\/td>\r\n<td>6<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td>8<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td>3<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>1<\/td>\r\n<td>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], we start from the inside with the input value 3. We then evaluate the inside expression [latex]g\\left(3\\right)[\/latex] using the table that defines the function [latex]g:[\/latex] [latex]g\\left(3\\right)=2[\/latex]. We can then use that result as the input to the function [latex]f[\/latex], so [latex]g\\left(3\\right)[\/latex] is replaced by 2 and we get [latex]f\\left(2\\right)[\/latex]. Then, using 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]\\begin{align}&amp;g\\left(3\\right)=2 \\\\[1.5mm]&amp; f\\left(g\\left(3\\right)\\right)=f\\left(2\\right)=8\\end{align}[\/latex]<\/p>\r\nTo evaluate [latex]g\\left(f\\left(3\\right)\\right)[\/latex], we first evaluate the inside expression [latex]f\\left(3\\right)[\/latex] using the first table: [latex]f\\left(3\\right)=3[\/latex]. Then, using the table for [latex]g[\/latex], we can evaluate\r\n<p style=\"text-align: center;\">[latex]g\\left(f\\left(3\\right)\\right)=g\\left(3\\right)=2[\/latex]<\/p>\r\nThe table below shows the composite functions [latex]f\\circ g[\/latex] and [latex]g\\circ f[\/latex] as tables.\r\n<table style=\"width: 30%;\" summary=\"Two rows and five columns. When x=3, g(3)=2, f(g(3))=8, f(3)=3, and g(f(3))=2.\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]g\\left(x\\right)[\/latex]<\/td>\r\n<td>[latex]f\\left(g\\left(x\\right)\\right)[\/latex]<\/td>\r\n<td>[latex]f\\left(x\\right)[\/latex]<\/td>\r\n<td>[latex]g\\left(f\\left(x\\right)\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td>2<\/td>\r\n<td>8<\/td>\r\n<td>3<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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=\"width: 30%;\" 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>[latex]x[\/latex]<\/th>\r\n<th>[latex]f\\left(x\\right)[\/latex]<\/th>\r\n<th>[latex]g\\left(x\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>1<\/td>\r\n<td>6<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td>8<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3<\/td>\r\n<td>3<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4<\/td>\r\n<td>1<\/td>\r\n<td>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[ohm_question height=\"300\"]3585[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Evaluating Composite Functions Using Graphs<\/h2>\r\nWhen we are given individual functions as graphs, the procedure for evaluating composite functions is similar to the process we use for evaluating tables. We read the input and output values, but this time, from the [latex]x\\text{-}[\/latex] and [latex]y\\text{-}[\/latex] axes of the graphs.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a composite function and graphs of its individual functions, evaluate it using the information provided by the graphs.<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Locate the given input to the inner function on the [latex]x\\text{-}[\/latex] axis of its graph.<\/li>\r\n \t<li>Read off the output of the inner function from the [latex]y\\text{-}[\/latex] axis of its graph.<\/li>\r\n \t<li>Locate the inner function output on the [latex]x\\text{-}[\/latex] axis of the graph of the outer function.<\/li>\r\n \t<li>Read the output of the outer function from the [latex]y\\text{-}[\/latex] axis of its graph. This is the output of the composite function.<\/li>\r\n<\/ol>\r\n<\/div>\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=\"Explanation of the composite function.\" 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 of a positive parabola (g(x)) and a negative parabola (f(x)). The following points are plotted: g(1)=3 and f(3)=6.\" width=\"975\" height=\"543\" \/>\r\n\r\nTo evaluate [latex]f\\left(g\\left(1\\right)\\right)[\/latex], we start with the inside evaluation.<span id=\"fs-id1165137644158\">\r\n<\/span>\r\n\r\nWe evaluate [latex]g\\left(1\\right)[\/latex] using the graph of [latex]g\\left(x\\right)[\/latex], 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]. We 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\nWe can then evaluate the composite function by looking to 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<h4>Analysis of the Solution<\/h4>\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 of a positive and negative parabola.\" width=\"975\" height=\"520\" \/>\r\n\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 graphs below, evaluate [latex]g\\left(f\\left(2\\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\/18195620\/CNX_Precalc_Figure_01_04_0042.jpg\" alt=\"Two graphs of a positive parabola (g(x)) and a negative parabola (f(x)). The following points are plotted: g(1)=3 and f(3)=6.\" width=\"975\" height=\"543\" \/>\r\n\r\n[reveal-answer q=\"682475\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"682475\"][latex]g\\left(f\\left(2\\right)\\right)=g\\left(5\\right)=3[\/latex][\/hidden-answer]\r\n\r\n[ohm_question height=\"300\"]69936[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Evaluating Composite Functions Using Formulas<\/h2>\r\nWhen evaluating a composite function where we have either created or been given formulas, the rule of working from the inside out remains the same. The input value to the outer function will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.\r\n\r\nWhile we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that will calculate the result of a composition [latex]f\\left(g\\left(x\\right)\\right)[\/latex]. To do this, we will extend our idea of function evaluation. Recall that, when we evaluate a function like [latex]f\\left(t\\right)={t}^{2}-t[\/latex], we substitute the value inside the parentheses into the formula wherever we see the input variable.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a formula for a composite function, evaluate the function.<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Evaluate the inside function using the input value or variable provided.<\/li>\r\n \t<li>Use the resulting output as the input to the outside function.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Evaluating a Composition of Functions Expressed as Formulas with a Numerical Input<\/h3>\r\nGiven [latex]f\\left(t\\right)={t}^{2}-{t}[\/latex] and [latex]h\\left(x\\right)=3x+2[\/latex], evaluate [latex]f\\left(h\\left(1\\right)\\right)[\/latex].\r\n\r\n[reveal-answer q=\"345593\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"345593\"]\r\n\r\nBecause the inside expression is [latex]h\\left(1\\right)[\/latex], we start by evaluating [latex]h\\left(x\\right)[\/latex] at 1.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}h\\left(1\\right)&amp;=3\\left(1\\right)+2\\\\[2mm] h\\left(1\\right)&amp;=5\\end{align}[\/latex]<\/p>\r\nThen [latex]f\\left(h\\left(1\\right)\\right)=f\\left(5\\right)[\/latex], so we evaluate [latex]f\\left(t\\right)[\/latex] at an input of 5.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}f\\left(h\\left(1\\right)\\right)&amp;=f\\left(5\\right)\\\\[2mm] f\\left(h\\left(1\\right)\\right)&amp;={5}^{2}-5\\\\[2mm] f\\left(h\\left(1\\right)\\right)&amp;=20\\end{align}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nIt makes no difference what the input variables [latex]t[\/latex] and [latex]x[\/latex] were called in this problem because we evaluated for specific numerical values.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGiven [latex]f\\left(t\\right)={t}^{2}-t[\/latex] and [latex]h\\left(x\\right)=3x+2[\/latex], evaluate\r\n\r\na.\u00a0 [latex]h\\left(f\\left(2\\right)\\right)[\/latex]\r\n\r\nb.\u00a0 [latex]h\\left(f\\left(-2\\right)\\right)[\/latex]\r\n\r\n[reveal-answer q=\"138476\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"138476\"]\r\n\r\na. 8; b. 20\r\n\r\nYou can check your work with an online graphing calculator. Enter the functions above into an online graphing calculator as they are defined. In the next line enter [latex]h\\left(f\\left(2\\right)\\right)[\/latex]. You should see [latex]=8[\/latex] in the bottom right corner.\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question]16852[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Evaluate a composition of functions using a table.<\/li>\n<li>Evaluate a composition of functions using an equation.<\/li>\n<\/ul>\n<\/div>\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 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: 30%;\" 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>[latex]x[\/latex]<\/th>\n<th>[latex]f\\left(x\\right)[\/latex]<\/th>\n<th>[latex]g\\left(x\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1<\/td>\n<td>6<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>8<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>3<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>1<\/td>\n<td>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], we start from the inside with the input value 3. We then evaluate the inside expression [latex]g\\left(3\\right)[\/latex] using the table that defines the function [latex]g:[\/latex] [latex]g\\left(3\\right)=2[\/latex]. We can then use that result as the input to the function [latex]f[\/latex], so [latex]g\\left(3\\right)[\/latex] is replaced by 2 and we get [latex]f\\left(2\\right)[\/latex]. Then, using 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]\\begin{align}&g\\left(3\\right)=2 \\\\[1.5mm]& f\\left(g\\left(3\\right)\\right)=f\\left(2\\right)=8\\end{align}[\/latex]<\/p>\n<p>To evaluate [latex]g\\left(f\\left(3\\right)\\right)[\/latex], we first evaluate the inside expression [latex]f\\left(3\\right)[\/latex] using the first table: [latex]f\\left(3\\right)=3[\/latex]. Then, using the table for [latex]g[\/latex], we can evaluate<\/p>\n<p style=\"text-align: center;\">[latex]g\\left(f\\left(3\\right)\\right)=g\\left(3\\right)=2[\/latex]<\/p>\n<p>The table below shows the composite functions [latex]f\\circ g[\/latex] and [latex]g\\circ f[\/latex] as tables.<\/p>\n<table style=\"width: 30%;\" summary=\"Two rows and five columns. When x=3, g(3)=2, f(g(3))=8, f(3)=3, and g(f(3))=2.\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]g\\left(x\\right)[\/latex]<\/td>\n<td>[latex]f\\left(g\\left(x\\right)\\right)[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)[\/latex]<\/td>\n<td>[latex]g\\left(f\\left(x\\right)\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>2<\/td>\n<td>8<\/td>\n<td>3<\/td>\n<td>2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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=\"width: 30%;\" 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>[latex]x[\/latex]<\/th>\n<th>[latex]f\\left(x\\right)[\/latex]<\/th>\n<th>[latex]g\\left(x\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1<\/td>\n<td>6<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>8<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>3<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>1<\/td>\n<td>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<p><iframe loading=\"lazy\" id=\"ohm3585\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3585&theme=oea&iframe_resize_id=ohm3585&show_question_numbers\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<h2>Evaluating Composite Functions Using Graphs<\/h2>\n<p>When we are given individual functions as graphs, the procedure for evaluating composite functions is similar to the process we use for evaluating tables. We read the input and output values, but this time, from the [latex]x\\text{-}[\/latex] and [latex]y\\text{-}[\/latex] axes of the graphs.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a composite function and graphs of its individual functions, evaluate it using the information provided by the graphs.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Locate the given input to the inner function on the [latex]x\\text{-}[\/latex] axis of its graph.<\/li>\n<li>Read off the output of the inner function from the [latex]y\\text{-}[\/latex] axis of its graph.<\/li>\n<li>Locate the inner function output on the [latex]x\\text{-}[\/latex] axis of the graph of the outer function.<\/li>\n<li>Read the output of the outer function from the [latex]y\\text{-}[\/latex] axis of its graph. This is the output of the composite function.<\/li>\n<\/ol>\n<\/div>\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=\"Explanation of the composite function.\" 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 of a positive parabola (g(x)) and a negative parabola (f(x)). The following points are plotted: g(1)=3 and f(3)=6.\" width=\"975\" height=\"543\" \/><\/p>\n<p>To evaluate [latex]f\\left(g\\left(1\\right)\\right)[\/latex], we start with the inside evaluation.<span id=\"fs-id1165137644158\"><br \/>\n<\/span><\/p>\n<p>We evaluate [latex]g\\left(1\\right)[\/latex] using the graph of [latex]g\\left(x\\right)[\/latex], 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]. We 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>We can then evaluate the composite function by looking to 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<h4>Analysis of the Solution<\/h4>\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 of a positive and negative parabola.\" width=\"975\" height=\"520\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Using the graphs below, evaluate [latex]g\\left(f\\left(2\\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\/18195620\/CNX_Precalc_Figure_01_04_0042.jpg\" alt=\"Two graphs of a positive parabola (g(x)) and a negative parabola (f(x)). The following points are plotted: g(1)=3 and f(3)=6.\" width=\"975\" height=\"543\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q682475\">Show Solution<\/span><\/p>\n<div id=\"q682475\" class=\"hidden-answer\" style=\"display: none\">[latex]g\\left(f\\left(2\\right)\\right)=g\\left(5\\right)=3[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm69936\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=69936&theme=oea&iframe_resize_id=ohm69936&show_question_numbers\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\n<h2>Evaluating Composite Functions Using Formulas<\/h2>\n<p>When evaluating a composite function where we have either created or been given formulas, the rule of working from the inside out remains the same. The input value to the outer function will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.<\/p>\n<p>While we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that will calculate the result of a composition [latex]f\\left(g\\left(x\\right)\\right)[\/latex]. To do this, we will extend our idea of function evaluation. Recall that, when we evaluate a function like [latex]f\\left(t\\right)={t}^{2}-t[\/latex], we substitute the value inside the parentheses into the formula wherever we see the input variable.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a formula for a composite function, evaluate the function.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Evaluate the inside function using the input value or variable provided.<\/li>\n<li>Use the resulting output as the input to the outside function.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Evaluating a Composition of Functions Expressed as Formulas with a Numerical Input<\/h3>\n<p>Given [latex]f\\left(t\\right)={t}^{2}-{t}[\/latex] and [latex]h\\left(x\\right)=3x+2[\/latex], evaluate [latex]f\\left(h\\left(1\\right)\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q345593\">Show Solution<\/span><\/p>\n<div id=\"q345593\" class=\"hidden-answer\" style=\"display: none\">\n<p>Because the inside expression is [latex]h\\left(1\\right)[\/latex], we start by evaluating [latex]h\\left(x\\right)[\/latex] at 1.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}h\\left(1\\right)&=3\\left(1\\right)+2\\\\[2mm] h\\left(1\\right)&=5\\end{align}[\/latex]<\/p>\n<p>Then [latex]f\\left(h\\left(1\\right)\\right)=f\\left(5\\right)[\/latex], so we evaluate [latex]f\\left(t\\right)[\/latex] at an input of 5.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}f\\left(h\\left(1\\right)\\right)&=f\\left(5\\right)\\\\[2mm] f\\left(h\\left(1\\right)\\right)&={5}^{2}-5\\\\[2mm] f\\left(h\\left(1\\right)\\right)&=20\\end{align}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>It makes no difference what the input variables [latex]t[\/latex] and [latex]x[\/latex] were called in this problem because we evaluated for specific numerical values.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given [latex]f\\left(t\\right)={t}^{2}-t[\/latex] and [latex]h\\left(x\\right)=3x+2[\/latex], evaluate<\/p>\n<p>a.\u00a0 [latex]h\\left(f\\left(2\\right)\\right)[\/latex]<\/p>\n<p>b.\u00a0 [latex]h\\left(f\\left(-2\\right)\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q138476\">Show Solution<\/span><\/p>\n<div id=\"q138476\" class=\"hidden-answer\" style=\"display: none\">\n<p>a. 8; b. 20<\/p>\n<p>You can check your work with an online graphing calculator. Enter the functions above into an online graphing calculator as they are defined. In the next line enter [latex]h\\left(f\\left(2\\right)\\right)[\/latex]. You should see [latex]=8[\/latex] in the bottom right corner.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm16852\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=16852&theme=oea&iframe_resize_id=ohm16852&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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-146\">\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>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>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 3585. <strong>Authored by<\/strong>: Reidel,Jessica. <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>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 69936. <strong>Authored by<\/strong>: Smart, Jim. <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>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 16852. <strong>Authored by<\/strong>: Pierpoint,William. <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>: IMathAS Community License CC-BY + GPL<\/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":395986,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College 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