{"id":47,"date":"2017-08-23T07:11:03","date_gmt":"2017-08-23T07:11:03","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=47"},"modified":"2017-09-21T19:43:33","modified_gmt":"2017-09-21T19:43:33","slug":"compositions-inverse-functions","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/compositions-inverse-functions\/","title":{"rendered":"Compositions &#038; Inverse Functions"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/solving-sytems-of-equations\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-i\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/sequences-series\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Compositions &amp; Inverse Functions<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will cover the definitions and uses of composite and inverse functions.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>A <em><strong>relation <\/strong><\/em>is\u00a0a set of ordered pairs. The set of values for the first\u00a0coordinate (<em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">)<\/span> is called the <strong><em>domain<\/em><\/strong>.\u00a0The set of values\u00a0for the second coordinate (<em><span style=\"text-decoration: none;\">y<\/span><\/em>)\u00a0is called the <em><strong>range<\/strong><\/em>.<\/li>\n<li>A <strong><em>function<\/em><\/strong> is a relation where\u00a0every value in the domain is paired with exactly one value in the\u00a0range. Functions often have an <strong><em>independent variable<\/em><\/strong> and a <em><strong>dependent\u00a0variable<\/strong><\/em>.<\/li>\n<\/ul>\n<section>\n<h3><strong>What are composite functions?<\/strong><\/h3>\n<p>The fact that a variable can be a function of a function leads us to the concept of <abbr title=\"a function of a function; given two functions f(x) and g(x), the composite function is denoted as (f\u00b7g)(x) or f(g(x)).\">composite\u00a0functions<\/abbr>. Suppose, for example, that Doug is selling\u00a0magazines from door to door. He earns a 10% commission on every\u00a0subscription he sells. By subscribing for a year, the customer\u00a0saves 50% off the retail price. If a customer subscribes to a\u00a0magazine that usually sells for $60 per year, what will Doug\u2019s\u00a0commission be?<\/p>\n<p>To find out, we use a chain of functions. Doug\u2019s\u00a0commission is a function of the subscription selling price, which\u00a0in turn is a function of the retail price. Let Doug\u2019s\u00a0commission equal DC, let S equal the subscription price, and R\u00a0equal the retail price.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s2_p2_clip_image002.gif\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/><\/center>This is a <em>composite function<\/em>. A composite function is\u00a0a way to combine functions by applying them in a specific order.\u00a0In function notation, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/s5_p2_html_3ac65648.gif\" width=\"128\" height=\"21\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>In the example, R = $60. Doug\u2019s commission = (\u0192\u03bf <em><span style=\"text-decoration: none;\">g) <\/span><\/em><span style=\"text-decoration: none;\">(<\/span><em><span style=\"text-decoration: none;\">R<\/span><\/em><span style=\"text-decoration: none;\">).\u00a0Solve for DC.<\/span><\/p>\n<p><center><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/s5_p2_html_42af1e35.gif\" width=\"115\" height=\"115\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/><\/center>Doug\u2019s commission is $3.00 for selling a subscription to\u00a0a magazine that usually sells for $60.00.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Let\u00a0<img decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s2_p2_clip_image010.gif\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/> and\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s2_p2_clip_image012.gif\" width=\"74\" height=\"14\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Which choice shows <img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/s5_p2_html_3b9555e3.gif\" width=\"77\" height=\"27\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<ol>\n<li><span style=\"text-decoration: none;\">2<\/span><em><span style=\"text-decoration: none;\"><em>x<\/em> <\/span><\/em><span style=\"text-decoration: none;\">+ 1<\/span><\/li>\n<li><span style=\"text-decoration: none;\">2<\/span><em><span style=\"text-decoration: none;\">x <\/span><\/em><span style=\"text-decoration: none;\">+ 4<\/span><\/li>\n<li><span style=\"text-decoration: none;\">2<\/span><em><span style=\"text-decoration: none;\">x <\/span><\/em><span style=\"text-decoration: none;\">\u2013 1<\/span><\/li>\n<li><em><span style=\"text-decoration: none;\">x <\/span><\/em><span style=\"text-decoration: none;\">\u2013 4<\/span><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is B. You know that <img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s2_p2_clip_image015.gif\" width=\"74\" height=\"14\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0To find <img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/s5_p2_html_3b9555e3.gif\" width=\"77\" height=\"27\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0substitute <img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s2_p2_clip_image019.gif\" width=\"29\" height=\"11\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/> for <em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\"> in \u0192(<\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">). <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s2_p2_clip_image021.gif\" width=\"149\" height=\"107\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Let <img decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s2_p2_clip_image023.gif\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/> and <img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s2_p2_clip_image025.gif\" width=\"74\" height=\"14\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Which choice shows <img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s2_p2_clip_image027.gif\" width=\"50\" height=\"14\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<ol>\n<li>\u20132<\/li>\n<li>\u20135<\/li>\n<li>5<\/li>\n<li>26<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is D. First, find <img decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s2_p2_clip_image029.gif\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s2_p2_clip_image031.gif\" width=\"74\" height=\"59\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p>The substitute \u20135 for <em>x <\/em>in <img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s2_p2_clip_image033.gif\" width=\"30\" height=\"14\" name=\"graphics19\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s2_p2_clip_image035.gif\" width=\"82\" height=\"62\" name=\"graphics20\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<h3><strong>What are inverse functions?<\/strong><\/h3>\n<p>You know that a relation is a set of ordered pairs. If we\u00a0exchange the domain for the range of the relation, we create a new\u00a0relation called the <abbr title=\"relation created by exchanging the domain for the range of the relation.\">inverse\u00a0relation<\/abbr>. If, when we create the inverse relation of a function, that inverse relation is also a function, it is called\u00a0an <abbr title=\"function that is derived from a given function by interchanging the two variables.\">inverse\u00a0function<\/abbr>. We notate the inverse function for <img decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s2_p3_clip_image002.gif\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/> with <img loading=\"lazy\" decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s2_p3_clip_image004.gif\" width=\"41\" height=\"17\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>Recall the earlier example of children in preschool and their\u00a0ages.<\/p>\n<h3>Children in Preschool<\/h3>\n<table width=\"200px\">\n<thead>\n<tr>\n<th>Child&#8217;s name<\/th>\n<th>Age<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Mary<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>Ashley<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>Brian<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>Charlie<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>Katie<\/td>\n<td>3.5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>As long as the children constituted the domain and their ages\u00a0constituted the range, the relation was a function. Mapping the\u00a0ordered pairs of the children and their ages in the opposite\u00a0direction, however, did not produce a function. Both Mary and\u00a0Charlie were paired with 3. If three was a member of the domain\u00a0rather than the range, it would have two members of the range\u00a0paired with it. Therefore, interchanging the domain and the range\u00a0would result in another relation, but not a function. All relations have inverse relations. Not all functions, however, have\u00a0inverse functions.<\/p>\n<p>The ordered pairs of an\u00a0inverse relation can be found by interchanging the <em>x-<\/em> and <em>y-<\/em>values\u00a0in the pairs. The inverse relation of {(1, 2) (3, 4) (5, 6) (7, 8)}\u00a0is {(2, 1) (4, 3) (6, 5) (8, 7)}. Graphically, a relation and its\u00a0inverse will produce a reflection across the line <em>y<\/em> = <em>x<\/em>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/Math%20Mod%202%5B1%5D.1%20Art%20005.JPG\" width=\"309\" height=\"315\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p>Find the inverse of a function that is an equation by\u00a0exchanging the variables.<\/p>\n<p>For example, let <img decoding=\"async\" class=\"no_margin non_block_image\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s2_p3_clip_image006.gif\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Substitute <em>y<\/em> for f(<em>x<\/em>), exchange variables, and solve\u00a0for <em>y<\/em>.<\/p>\n<p align=\"CENTER\"><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s2_p3_clip_image008.gif\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p>If we have found the\u00a0inverse, it will undo the original function. That is, if \u0192(<em>x<\/em>)\u00a0= <em>y<\/em>,\u00a0then \u0192<sup>\u20131<\/sup>(<em>y<\/em>)\u00a0= <em>x<\/em>.\u00a0Use 6 to check our work.<\/p>\n<p align=\"CENTER\"><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s2_p3_clip_image010.gif\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<p>If the functions are inverses, then \u0192 <sup>\u20131<\/sup>(22)\u00a0should equal 6.<\/p>\n<p align=\"CENTER\"><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/2\/images2\/s2_p3_clip_image012.gif\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/solving-sytems-of-equations\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/algebra-functions-i\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/sequences-series\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Compositions &amp; Inverse Functions Objective In this lesson, you will cover the definitions and uses of composite and inverse functions. Previously Covered: A relation is\u00a0a set of ordered pairs. The set of values for the first\u00a0coordinate (x) is called the domain.\u00a0The set of values\u00a0for the second coordinate (y)\u00a0is called [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-47","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/47","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=47"}],"version-history":[{"count":7,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/47\/revisions"}],"predecessor-version":[{"id":788,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/47\/revisions\/788"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=47"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}