{"id":1290,"date":"2025-07-24T04:04:44","date_gmt":"2025-07-24T04:04:44","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1290"},"modified":"2026-03-18T02:47:44","modified_gmt":"2026-03-18T02:47:44","slug":"composition-of-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/composition-of-functions-fresh-take\/","title":{"raw":"Composition of Functions: Fresh Take","rendered":"Composition of Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Combine and evaluate functions using algebraic operations.<\/li>\r\n \t<li>Create and evaluate a new function by composition of functions.<\/li>\r\n \t<li>Find the domain of a composite function.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Combining Functions Using Algebraic Operations<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Function Addition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex](f + g)(x) = f(x) + g(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Function Subtraction:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex](f - g)(x) = f(x) - g(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Function Multiplication:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex](f \\cdot g)(x) = f(x) \\cdot g(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Function Division:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\left(\\frac{f}{g}\\right)(x) = \\frac{f(x)}{g(x)}[\/latex], where [latex]g(x) \\neq 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Domain Considerations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Resulting function's domain may be restricted, especially for division<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find and simplify the functions [latex]\\left(f\\cdot{g}\\right)\\left(x\\right)[\/latex] and [latex]\\left(f-g\\right)\\left(x\\right)[\/latex].\r\n<div id=\"fs-id1165137434994\" class=\"equation unnumbered\">[latex]f\\left(x\\right)=x - 1\\text{ and }g\\left(x\\right)={x}^{2}-1[\/latex]<\/div>\r\nAre they the same function?\r\n[reveal-answer q=\"721147\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"721147\"][latex]\\left(f\\cdot{g}\\right)\\left(x\\right)=f\\left(x\\right)g\\left(x\\right)=\\left(x - 1\\right)\\left({x}^{2}-1\\right)={x}^{3}-{x}^{2}-x+1[\/latex]\r\n\r\n[latex]\\left(f-g\\right)\\left(x\\right)=f\\left(x\\right)-g\\left(x\\right)=\\left(x - 1\\right)-\\left({x}^{2}-1\\right)=x-{x}^{2}[\/latex]\r\n\r\nNo, the functions are not the same.[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Create a Function by Composition of Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex](f \\circ g)(x) = f(g(x))[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Read as \"f composed with g of x\" or \"f of g of x\"<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Order Matters:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Generally, [latex](f \\circ g)(x) \\neq (g \\circ f)(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Domain Considerations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Domain of [latex]f \\circ g[\/latex]: All [latex]x[\/latex] in domain of [latex]g[\/latex] where [latex]g(x)[\/latex] is in domain of [latex]f[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Evaluation Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Work from innermost function outward<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Not Multiplication:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex](f \\circ g)(x) \\neq (f \\cdot g)(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">The gravitational force on a planet a distance [latex]r[\/latex] from the sun is given by the function [latex]G\\left(r\\right)[\/latex]. The acceleration of a planet subjected to any force [latex]F[\/latex] is given by the function [latex]a\\left(F\\right)[\/latex]. Form a meaningful composition of these two functions, and explain what it means.\r\n[reveal-answer q=\"349230\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"349230\"]A gravitational force is still a force, so [latex]a\\left(G\\left(r\\right)\\right)[\/latex] makes sense as the acceleration of a planet at a distance r from the Sun (due to gravity), but [latex]G\\left(a\\left(F\\right)\\right)[\/latex] does not make sense.[\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Using the functions provided, find [latex]f\\left(g\\left(x\\right)\\right)[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)[\/latex].\r\n[latex]f\\left(x\\right)=2x+1\\\\g\\left(x\\right)=3-x[\/latex]\r\n[reveal-answer q=\"822785\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"822785\"]Let\u2019s begin by substituting [latex]g\\left(x\\right)[\/latex] into [latex]f\\left(x\\right)[\/latex].[latex]\\begin{align}f\\left(g\\left(x\\right)\\right)&amp;=2\\left(3-x\\right)+1 \\\\[2mm] &amp;=6 - 2x+1 \\\\[2mm] &amp;=7 - 2x \\end{align}[\/latex]Now we can substitute [latex]f\\left(x\\right)[\/latex] into [latex]g\\left(x\\right)[\/latex].[latex]\\begin{align}g\\left(f\\left(x\\right)\\right)&amp;=3-\\left(2x+1\\right) \\\\[2mm] &amp;=3 - 2x - 1 \\\\[2mm] &amp;=-2x+2 \\end{align}[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">\r\n<h2><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script><\/h2>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ebdeagdb-qxBmISCJSME\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/qxBmISCJSME?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ebdeagdb-qxBmISCJSME\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844424&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ebdeagdb-qxBmISCJSME&amp;vembed=0&amp;video_id=qxBmISCJSME&amp;video_target=tpm-plugin-ebdeagdb-qxBmISCJSME\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Composite+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cComposite Functions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">\r\n<h2><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script><\/h2>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fhbccgdf-yQ6BPmRTzBQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/yQ6BPmRTzBQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fhbccgdf-yQ6BPmRTzBQ\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844425&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fhbccgdf-yQ6BPmRTzBQ&amp;vembed=0&amp;video_id=yQ6BPmRTzBQ&amp;video_target=tpm-plugin-fhbccgdf-yQ6BPmRTzBQ\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+4+-+Domain+of+a+Composite+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 4: Domain of a Composite Function\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Evaluating Composite Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">General Approach:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Work from inside to outside<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Evaluate inner function first, use result as input for outer function<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Using Tables:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Read input\/output values directly from table entries<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex](f \\circ g)(x) = f(g(x))[\/latex]: Look up [latex]g(x)[\/latex], then use result to find [latex]f[\/latex] value<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Using Graphs:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Read input\/output values from [latex]x[\/latex] and [latex]y[\/latex] axes<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Follow points across graphs for composite functions<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Using Formulas:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Substitute expression for inner function into outer function<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify to get formula for composite function<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Domain Considerations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Check domains of both inner and outer functions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Ensure composition is valid for given input<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Using 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<\/section><section class=\"textbox example\" aria-label=\"Example\">Using the graphs below, evaluate [latex]g\\left(f\\left(2\\right)\\right)[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img 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\" \/> Graphs of g(x) and f(x) with points plotted[\/caption]\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<\/section><section class=\"textbox example\" aria-label=\"Example\">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].[reveal-answer q=\"345593\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"345593\"]Because 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<strong>Analysis of the Solution<\/strong>\r\n\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<\/section><section class=\"textbox example\" aria-label=\"Example\">Given [latex]f\\left(t\\right)={t}^{2}-t[\/latex] and [latex]h\\left(x\\right)=3x+2[\/latex], evaluatea.\u00a0 [latex]h\\left(f\\left(2\\right)\\right)[\/latex]b.\u00a0 [latex]h\\left(f\\left(-2\\right)\\right)[\/latex][reveal-answer q=\"138476\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"138476\"]a. 8; b. 20You 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.[\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">\r\n<h2><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script><\/h2>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ggeaedcg-b-i7N0hE-Ys\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/b-i7N0hE-Ys?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ggeaedcg-b-i7N0hE-Ys\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844426&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ggeaedcg-b-i7N0hE-Ys&amp;vembed=0&amp;video_id=b-i7N0hE-Ys&amp;video_target=tpm-plugin-ggeaedcg-b-i7N0hE-Ys\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Evaluate+Composite+Functions+from+Graphs_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Evaluate Composite Functions from Graphs\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dgafdbbb-y2kJI9XnyLY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/y2kJI9XnyLY?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dgafdbbb-y2kJI9XnyLY\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844427&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dgafdbbb-y2kJI9XnyLY&amp;vembed=0&amp;video_id=y2kJI9XnyLY&amp;video_target=tpm-plugin-dgafdbbb-y2kJI9XnyLY\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Composite+Function+Values_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Composite Function Values\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Decomposing a Composite Function<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Breaking down a complex function into simpler component functions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]f(x) = g(h(x))[\/latex], where [latex]g[\/latex] and [latex]h[\/latex] are simpler functions<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Multiple Solutions:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">There's often more than one way to decompose a function<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Choose the decomposition that seems most useful or intuitive<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify a \"function inside a function\" in the original expression<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Define inner function [latex]h(x)[\/latex] and outer function [latex]g(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verification:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Recompose the functions to check if [latex]g(h(x))[\/latex] equals the original function<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Applications:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Simplifying complex functions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Understanding function structure<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solving certain types of equations<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Write [latex]f\\left(x\\right)=\\dfrac{4}{3-\\sqrt{4+{x}^{2}}}[\/latex] as the composition of two functions.[reveal-answer q=\"489928\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"489928\"]Possible answer:\r\n<p id=\"fs-id1165135333608\">[latex]g\\left(x\\right)=\\sqrt{4+{x}^{2}}[\/latex]<\/p>\r\n[latex]h\\left(x\\right)=\\dfrac{4}{3-x}[\/latex]\r\n\r\n[latex]f=h\\circ g[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api \" type=\"text\/javascript\"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dafbahhh-gFSSk8jaAwA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/gFSSk8jaAwA?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dafbahhh-gFSSk8jaAwA\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844428&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dafbahhh-gFSSk8jaAwA&amp;vembed=0&amp;video_id=gFSSk8jaAwA&amp;video_target=tpm-plugin-dafbahhh-gFSSk8jaAwA\" type=\"text\/javascript\"><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Decompose+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Decompose Functions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Combine and evaluate functions using algebraic operations.<\/li>\n<li>Create and evaluate a new function by composition of functions.<\/li>\n<li>Find the domain of a composite function.<\/li>\n<\/ul>\n<\/section>\n<h2>Combining Functions Using Algebraic Operations<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Function Addition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex](f + g)(x) = f(x) + g(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Function Subtraction:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex](f - g)(x) = f(x) - g(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Function Multiplication:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex](f \\cdot g)(x) = f(x) \\cdot g(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Function Division:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\left(\\frac{f}{g}\\right)(x) = \\frac{f(x)}{g(x)}[\/latex], where [latex]g(x) \\neq 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Domain Considerations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Resulting function&#8217;s domain may be restricted, especially for division<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find and simplify the functions [latex]\\left(f\\cdot{g}\\right)\\left(x\\right)[\/latex] and [latex]\\left(f-g\\right)\\left(x\\right)[\/latex].<\/p>\n<div id=\"fs-id1165137434994\" class=\"equation unnumbered\">[latex]f\\left(x\\right)=x - 1\\text{ and }g\\left(x\\right)={x}^{2}-1[\/latex]<\/div>\n<p>Are they the same function?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q721147\">Show Solution<\/button><\/p>\n<div id=\"q721147\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(f\\cdot{g}\\right)\\left(x\\right)=f\\left(x\\right)g\\left(x\\right)=\\left(x - 1\\right)\\left({x}^{2}-1\\right)={x}^{3}-{x}^{2}-x+1[\/latex]<\/p>\n<p>[latex]\\left(f-g\\right)\\left(x\\right)=f\\left(x\\right)-g\\left(x\\right)=\\left(x - 1\\right)-\\left({x}^{2}-1\\right)=x-{x}^{2}[\/latex]<\/p>\n<p>No, the functions are not the same.<\/p><\/div>\n<\/div>\n<\/section>\n<h2>Create a Function by Composition of Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex](f \\circ g)(x) = f(g(x))[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Read as &#8220;f composed with g of x&#8221; or &#8220;f of g of x&#8221;<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Order Matters:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Generally, [latex](f \\circ g)(x) \\neq (g \\circ f)(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Domain Considerations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Domain of [latex]f \\circ g[\/latex]: All [latex]x[\/latex] in domain of [latex]g[\/latex] where [latex]g(x)[\/latex] is in domain of [latex]f[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Evaluation Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Work from innermost function outward<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Not Multiplication:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex](f \\circ g)(x) \\neq (f \\cdot g)(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">The gravitational force on a planet a distance [latex]r[\/latex] from the sun is given by the function [latex]G\\left(r\\right)[\/latex]. The acceleration of a planet subjected to any force [latex]F[\/latex] is given by the function [latex]a\\left(F\\right)[\/latex]. Form a meaningful composition of these two functions, and explain what it means.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q349230\">Show Solution<\/button><\/p>\n<div id=\"q349230\" class=\"hidden-answer\" style=\"display: none\">A gravitational force is still a force, so [latex]a\\left(G\\left(r\\right)\\right)[\/latex] makes sense as the acceleration of a planet at a distance r from the Sun (due to gravity), but [latex]G\\left(a\\left(F\\right)\\right)[\/latex] does not make sense.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Using the functions provided, find [latex]f\\left(g\\left(x\\right)\\right)[\/latex] and [latex]g\\left(f\\left(x\\right)\\right)[\/latex].<br \/>\n[latex]f\\left(x\\right)=2x+1\\\\g\\left(x\\right)=3-x[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q822785\">Show Solution<\/button><\/p>\n<div id=\"q822785\" class=\"hidden-answer\" style=\"display: none\">Let\u2019s begin by substituting [latex]g\\left(x\\right)[\/latex] into [latex]f\\left(x\\right)[\/latex].[latex]\\begin{align}f\\left(g\\left(x\\right)\\right)&=2\\left(3-x\\right)+1 \\\\[2mm] &=6 - 2x+1 \\\\[2mm] &=7 - 2x \\end{align}[\/latex]Now we can substitute [latex]f\\left(x\\right)[\/latex] into [latex]g\\left(x\\right)[\/latex].[latex]\\begin{align}g\\left(f\\left(x\\right)\\right)&=3-\\left(2x+1\\right) \\\\[2mm] &=3 - 2x - 1 \\\\[2mm] &=-2x+2 \\end{align}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\n<h2><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/h2>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ebdeagdb-qxBmISCJSME\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/qxBmISCJSME?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ebdeagdb-qxBmISCJSME\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844424&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ebdeagdb-qxBmISCJSME&amp;vembed=0&amp;video_id=qxBmISCJSME&amp;video_target=tpm-plugin-ebdeagdb-qxBmISCJSME\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Composite+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cComposite Functions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\n<h2><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/h2>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fhbccgdf-yQ6BPmRTzBQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/yQ6BPmRTzBQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fhbccgdf-yQ6BPmRTzBQ\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844425&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-fhbccgdf-yQ6BPmRTzBQ&amp;vembed=0&amp;video_id=yQ6BPmRTzBQ&amp;video_target=tpm-plugin-fhbccgdf-yQ6BPmRTzBQ\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+4+-+Domain+of+a+Composite+Function_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 4: Domain of a Composite Function\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Evaluating Composite Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">General Approach:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Work from inside to outside<\/li>\n<li class=\"whitespace-normal break-words\">Evaluate inner function first, use result as input for outer function<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Using Tables:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Read input\/output values directly from table entries<\/li>\n<li class=\"whitespace-normal break-words\">[latex](f \\circ g)(x) = f(g(x))[\/latex]: Look up [latex]g(x)[\/latex], then use result to find [latex]f[\/latex] value<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Using Graphs:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Read input\/output values from [latex]x[\/latex] and [latex]y[\/latex] axes<\/li>\n<li class=\"whitespace-normal break-words\">Follow points across graphs for composite functions<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Using Formulas:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Substitute expression for inner function into outer function<\/li>\n<li class=\"whitespace-normal break-words\">Simplify to get formula for composite function<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Domain Considerations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Check domains of both inner and outer functions<\/li>\n<li class=\"whitespace-normal break-words\">Ensure composition is valid for given input<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q161706\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Using the graphs below, evaluate [latex]g\\left(f\\left(2\\right)\\right)[\/latex].<\/p>\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" 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\" \/><figcaption class=\"wp-caption-text\">Graphs of g(x) and f(x) with points plotted<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q682475\">Show Solution<\/button><\/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<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q345593\">Show Solution<\/button><\/p>\n<div id=\"q345593\" class=\"hidden-answer\" style=\"display: none\">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<p><strong>Analysis of the Solution<\/strong><\/p>\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<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Given [latex]f\\left(t\\right)={t}^{2}-t[\/latex] and [latex]h\\left(x\\right)=3x+2[\/latex], evaluatea.\u00a0 [latex]h\\left(f\\left(2\\right)\\right)[\/latex]b.\u00a0 [latex]h\\left(f\\left(-2\\right)\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q138476\">Show Solution<\/button><\/p>\n<div id=\"q138476\" class=\"hidden-answer\" style=\"display: none\">a. 8; b. 20You 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.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\n<h2><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/h2>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ggeaedcg-b-i7N0hE-Ys\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/b-i7N0hE-Ys?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ggeaedcg-b-i7N0hE-Ys\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844426&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-ggeaedcg-b-i7N0hE-Ys&amp;vembed=0&amp;video_id=b-i7N0hE-Ys&amp;video_target=tpm-plugin-ggeaedcg-b-i7N0hE-Ys\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Evaluate+Composite+Functions+from+Graphs_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Evaluate Composite Functions from Graphs\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dgafdbbb-y2kJI9XnyLY\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/y2kJI9XnyLY?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dgafdbbb-y2kJI9XnyLY\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844427&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dgafdbbb-y2kJI9XnyLY&amp;vembed=0&amp;video_id=y2kJI9XnyLY&amp;video_target=tpm-plugin-dgafdbbb-y2kJI9XnyLY\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Composite+Function+Values_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Composite Function Values\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Decomposing a Composite Function<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Breaking down a complex function into simpler component functions<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f(x) = g(h(x))[\/latex], where [latex]g[\/latex] and [latex]h[\/latex] are simpler functions<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Multiple Solutions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">There&#8217;s often more than one way to decompose a function<\/li>\n<li class=\"whitespace-normal break-words\">Choose the decomposition that seems most useful or intuitive<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify a &#8220;function inside a function&#8221; in the original expression<\/li>\n<li class=\"whitespace-normal break-words\">Define inner function [latex]h(x)[\/latex] and outer function [latex]g(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Verification:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Recompose the functions to check if [latex]g(h(x))[\/latex] equals the original function<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Simplifying complex functions<\/li>\n<li class=\"whitespace-normal break-words\">Understanding function structure<\/li>\n<li class=\"whitespace-normal break-words\">Solving certain types of equations<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Write [latex]f\\left(x\\right)=\\dfrac{4}{3-\\sqrt{4+{x}^{2}}}[\/latex] as the composition of two functions.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q489928\">Show Solution<\/button><\/p>\n<div id=\"q489928\" class=\"hidden-answer\" style=\"display: none\">Possible answer:<\/p>\n<p id=\"fs-id1165135333608\">[latex]g\\left(x\\right)=\\sqrt{4+{x}^{2}}[\/latex]<\/p>\n<p>[latex]h\\left(x\\right)=\\dfrac{4}{3-x}[\/latex]<\/p>\n<p>[latex]f=h\\circ g[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script src=\"https:\/\/www.youtube.com\/iframe_api\" type=\"text\/javascript\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dafbahhh-gFSSk8jaAwA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/gFSSk8jaAwA?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dafbahhh-gFSSk8jaAwA\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&amp;cc_minimizable=1&amp;cc_minimize_on_load=0&amp;cc_multi_text_track=0&amp;cc_overlay=1&amp;cc_searchable=0&amp;embed=ajax&amp;mf=12844428&amp;p3sdk_version=1.11.7&amp;p=20361&amp;player_type=youtube&amp;plugin_skin=dark&amp;target=3p-plugin-target-dafbahhh-gFSSk8jaAwA&amp;vembed=0&amp;video_id=gFSSk8jaAwA&amp;video_target=tpm-plugin-dafbahhh-gFSSk8jaAwA\" type=\"text\/javascript\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Decompose+Functions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Decompose Functions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Composite Functions\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/qxBmISCJSME\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 4: Domain of a Composite Function\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/yQ6BPmRTzBQ\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Evaluate Composite Functions from Graphs\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/b-i7N0hE-Ys\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 1: Composite Function 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