{"id":148,"date":"2023-06-21T13:22:38","date_gmt":"2023-06-21T13:22:38","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-compositions-of-functions\/"},"modified":"2023-07-03T20:51:11","modified_gmt":"2023-07-03T20:51:11","slug":"summary-compositions-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-compositions-of-functions\/","title":{"raw":"Summary: Compositions of Functions","rendered":"Summary: Compositions of Functions"},"content":{"raw":"\n\n<h2>Key Equation<\/h2>\n<section id=\"fs-id1165135388428\" class=\"key-equations\" data-depth=\"1\">\n<table id=\"eip-id1165134118229\" summary=\"..\"><colgroup> <col data-align=\"center\"> <col data-align=\"center\"><\/colgroup>\n<tbody>\n<tr>\n<td>Composite function<\/td>\n<td>[latex]\\left(f\\circ g\\right)\\left(x\\right)=f\\left(g\\left(x\\right)\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section><section id=\"fs-id1165137768015\" class=\"key-concepts\" data-depth=\"1\">\n<h2 data-type=\"title\">Key Concepts<\/h2>\n<ul id=\"fs-id1165137754842\">\n \t<li>We can perform algebraic operations on functions.<\/li>\n \t<li>When functions are combined, the output of the first (inner) function becomes the input of the second (outer) function.<\/li>\n \t<li>The function produced by combining two functions is a composite function.<\/li>\n \t<li>The order of function composition must be considered when interpreting the meaning of composite functions.<\/li>\n \t<li>A composite function can be evaluated by evaluating the inner function using the given input value and then evaluating the outer function taking as its input the output of the inner function.<\/li>\n \t<li>A composite function can be evaluated from a table.<\/li>\n \t<li>A composite function can be evaluated from a graph.<\/li>\n \t<li>A composite function can be evaluated from a formula.<\/li>\n \t<li>The domain of a composite function consists of those inputs in the domain of the inner function that correspond to outputs of the inner function that are in the domain of the outer function.<\/li>\n \t<li>Just as functions can be combined to form a composite function, composite functions can be decomposed into simpler functions.<\/li>\n \t<li>Functions can often be decomposed in more than one way.<\/li>\n<\/ul>\n<\/section><section id=\"fs-id1165137611819\" class=\"section-exercises\" data-depth=\"1\"><section class=\"section-exercises\" data-depth=\"1\"><section class=\"section-exercises\" data-depth=\"1\">\n<div data-type=\"glossary\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl id=\"fs-id1165137832421\" class=\"definition\">\n \t<dt><strong>composite function<\/strong><\/dt>\n \t<dd id=\"fs-id1165137832426\">the new function formed by function composition, when the output of one function is used as the input of another<\/dd>\n<\/dl>\n<\/div>\n<\/section><\/section><\/section>\n\n","rendered":"<h2>Key Equation<\/h2>\n<section id=\"fs-id1165135388428\" class=\"key-equations\" data-depth=\"1\">\n<table id=\"eip-id1165134118229\" summary=\"..\">\n<colgroup>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/><\/colgroup>\n<tbody>\n<tr>\n<td>Composite function<\/td>\n<td>[latex]\\left(f\\circ g\\right)\\left(x\\right)=f\\left(g\\left(x\\right)\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165137768015\" class=\"key-concepts\" data-depth=\"1\">\n<h2 data-type=\"title\">Key Concepts<\/h2>\n<ul id=\"fs-id1165137754842\">\n<li>We can perform algebraic operations on functions.<\/li>\n<li>When functions are combined, the output of the first (inner) function becomes the input of the second (outer) function.<\/li>\n<li>The function produced by combining two functions is a composite function.<\/li>\n<li>The order of function composition must be considered when interpreting the meaning of composite functions.<\/li>\n<li>A composite function can be evaluated by evaluating the inner function using the given input value and then evaluating the outer function taking as its input the output of the inner function.<\/li>\n<li>A composite function can be evaluated from a table.<\/li>\n<li>A composite function can be evaluated from a graph.<\/li>\n<li>A composite function can be evaluated from a formula.<\/li>\n<li>The domain of a composite function consists of those inputs in the domain of the inner function that correspond to outputs of the inner function that are in the domain of the outer function.<\/li>\n<li>Just as functions can be combined to form a composite function, composite functions can be decomposed into simpler functions.<\/li>\n<li>Functions can often be decomposed in more than one way.<\/li>\n<\/ul>\n<\/section>\n<section id=\"fs-id1165137611819\" class=\"section-exercises\" data-depth=\"1\">\n<section class=\"section-exercises\" data-depth=\"1\">\n<section class=\"section-exercises\" data-depth=\"1\">\n<div data-type=\"glossary\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl id=\"fs-id1165137832421\" class=\"definition\">\n<dt><strong>composite function<\/strong><\/dt>\n<dd id=\"fs-id1165137832426\">the new function formed by function composition, when the output of one function is used as the input of another<\/dd>\n<\/dl>\n<\/div>\n<\/section>\n<\/section>\n<\/section>\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-148\">\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><\/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":10,"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 Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at 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