{"id":102,"date":"2021-02-03T21:02:55","date_gmt":"2021-02-03T21:02:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=102"},"modified":"2021-03-15T20:00:27","modified_gmt":"2021-03-15T20:00:27","slug":"summary-of-review-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/summary-of-review-of-functions\/","title":{"raw":"Summary of Review of Functions","rendered":"Summary of Review of Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1170572176922\">\r\n \t<li>A function is a mapping from a set of inputs to a set of outputs with exactly one output for each input.<\/li>\r\n \t<li>If no domain is stated for a function [latex]y=f(x)[\/latex], the domain is considered to be the set of all real numbers [latex]x[\/latex] for which the function is defined.<\/li>\r\n \t<li>When sketching the graph of a function [latex]f[\/latex], each vertical line may intersect the graph, at most, once.<\/li>\r\n \t<li>A function may have any number of zeros, but it has, at most, one [latex]y[\/latex]-intercept.<\/li>\r\n \t<li>To define the composition [latex]g\\circ f[\/latex], the range of [latex]f[\/latex] must be contained in the domain of [latex]g[\/latex].<\/li>\r\n \t<li>Even functions are symmetric about the [latex]y[\/latex]-axis whereas odd functions are symmetric about the origin.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1170572554776\">\r\n \t<li><strong>Composition of two functions<\/strong>\r\n[latex](g\\circ f)(x)=g(f(x))[\/latex]<\/li>\r\n \t<li><strong>Absolute value function<\/strong>\r\n[latex]f(x) = |x| = \\begin{cases} x, &amp; x \\ge 0 \\\\ -x, &amp; x &lt; 0 \\end{cases}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170572443492\" class=\"definition\">\r\n \t<dt>absolute value function<\/dt>\r\n \t<dd id=\"fs-id1170572443498\">[latex]f(x) = |x| = \\begin{cases} x, &amp; x \\ge 0 \\\\ -x, &amp; x &lt; 0 \\end{cases}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572443550\" class=\"definition\">\r\n \t<dt>composite function<\/dt>\r\n \t<dd id=\"fs-id1170572443555\">given two functions [latex]f[\/latex] and [latex]g[\/latex], a new function, denoted [latex]g\\circ f[\/latex], such that [latex](g\\circ f)(x)=g(f(x))[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572548156\" class=\"definition\">\r\n \t<dt>decreasing on the interval [latex]I[\/latex]<\/dt>\r\n \t<dd id=\"fs-id1170572548165\">a function decreasing on the interval [latex]I[\/latex] if, for all [latex]x_1, \\, x_2\\in I, \\, f(x_1)\\ge f(x_2)[\/latex] if [latex]x_1&lt;x_2[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573520938\" class=\"definition\">\r\n \t<dt>dependent variable<\/dt>\r\n \t<dd id=\"fs-id1170572548252\">the output variable for a function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572548257\" class=\"definition\">\r\n \t<dt>domain<\/dt>\r\n \t<dd id=\"fs-id1170572548262\">the set of inputs for a function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572548267\" class=\"definition\">\r\n \t<dt>even function<\/dt>\r\n \t<dd id=\"fs-id1170572548272\">a function is even if [latex]f(\u2212x)=f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572548320\" class=\"definition\">\r\n \t<dt>function<\/dt>\r\n \t<dd id=\"fs-id1170572549920\">a set of inputs, a set of outputs, and a rule for mapping each input to exactly one output<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572549925\" class=\"definition\">\r\n \t<dt>graph of a function<\/dt>\r\n \t<dd id=\"fs-id1170572549930\">the set of points [latex](x,y)[\/latex] such that [latex]x[\/latex] is in the domain of [latex]f[\/latex] and [latex]y=f(x)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572549980\" class=\"definition\">\r\n \t<dt>increasing on the interval [latex]I[\/latex]<\/dt>\r\n \t<dd id=\"fs-id1170572549988\">a function increasing on the interval [latex]I[\/latex] if for all [latex]x_1, \\, x_2\\in I, \\, f(x_1)\\le f(x_2)[\/latex] if [latex]x_1&lt;x_2[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572550070\" class=\"definition\">\r\n \t<dt>independent variable<\/dt>\r\n \t<dd id=\"fs-id1170572550076\">the input variable for a function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572550080\" class=\"definition\">\r\n \t<dt>odd function<\/dt>\r\n \t<dd id=\"fs-id1170572550086\">a function is odd if [latex]f(\u2212x)=\u2212f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572550135\" class=\"definition\">\r\n \t<dt>range<\/dt>\r\n \t<dd id=\"fs-id1170572235100\">the set of outputs for a function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572235105\" class=\"definition\">\r\n \t<dt>symmetry about the origin<\/dt>\r\n \t<dd id=\"fs-id1170572235110\">the graph of a function [latex]f[\/latex] is symmetric about the origin if [latex](\u2212x,\u2212y)[\/latex] is on the graph of [latex]f[\/latex] whenever [latex](x,y)[\/latex] is on the graph<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572235164\" class=\"definition\">\r\n \t<dt>symmetry about the [latex]y[\/latex]-axis<\/dt>\r\n \t<dd id=\"fs-id1170572235175\">the graph of a function [latex]f[\/latex] is symmetric about the [latex]y[\/latex]-axis if [latex](\u2212x,y)[\/latex] is on the graph of [latex]f[\/latex] whenever [latex](x,y)[\/latex] is on the graph<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572235230\" class=\"definition\">\r\n \t<dt>table of values<\/dt>\r\n \t<dd id=\"fs-id1170572235236\">a table containing a list of inputs and their corresponding outputs<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572235240\" class=\"definition\">\r\n \t<dt>vertical line test<\/dt>\r\n \t<dd id=\"fs-id1170572235245\">given the graph of a function, every vertical line intersects the graph, at most, once<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572235250\" class=\"definition\">\r\n \t<dt>zeros of a function<\/dt>\r\n \t<dd id=\"fs-id1170572235255\">when a real number [latex]x[\/latex] is a zero of a function [latex]f[\/latex], [latex]f(x)=0[\/latex]<\/dd>\r\n<\/dl>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1170572176922\">\n<li>A function is a mapping from a set of inputs to a set of outputs with exactly one output for each input.<\/li>\n<li>If no domain is stated for a function [latex]y=f(x)[\/latex], the domain is considered to be the set of all real numbers [latex]x[\/latex] for which the function is defined.<\/li>\n<li>When sketching the graph of a function [latex]f[\/latex], each vertical line may intersect the graph, at most, once.<\/li>\n<li>A function may have any number of zeros, but it has, at most, one [latex]y[\/latex]-intercept.<\/li>\n<li>To define the composition [latex]g\\circ f[\/latex], the range of [latex]f[\/latex] must be contained in the domain of [latex]g[\/latex].<\/li>\n<li>Even functions are symmetric about the [latex]y[\/latex]-axis whereas odd functions are symmetric about the origin.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170572554776\">\n<li><strong>Composition of two functions<\/strong><br \/>\n[latex](g\\circ f)(x)=g(f(x))[\/latex]<\/li>\n<li><strong>Absolute value function<\/strong><br \/>\n[latex]f(x) = |x| = \\begin{cases} x, & x \\ge 0 \\\\ -x, & x < 0 \\end{cases}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170572443492\" class=\"definition\">\n<dt>absolute value function<\/dt>\n<dd id=\"fs-id1170572443498\">[latex]f(x) = |x| = \\begin{cases} x, & x \\ge 0 \\\\ -x, & x < 0 \\end{cases}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572443550\" class=\"definition\">\n<dt>composite function<\/dt>\n<dd id=\"fs-id1170572443555\">given two functions [latex]f[\/latex] and [latex]g[\/latex], a new function, denoted [latex]g\\circ f[\/latex], such that [latex](g\\circ f)(x)=g(f(x))[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572548156\" class=\"definition\">\n<dt>decreasing on the interval [latex]I[\/latex]<\/dt>\n<dd id=\"fs-id1170572548165\">a function decreasing on the interval [latex]I[\/latex] if, for all [latex]x_1, \\, x_2\\in I, \\, f(x_1)\\ge f(x_2)[\/latex] if [latex]x_1<x_2[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170573520938\" class=\"definition\">\n<dt>dependent variable<\/dt>\n<dd id=\"fs-id1170572548252\">the output variable for a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572548257\" class=\"definition\">\n<dt>domain<\/dt>\n<dd id=\"fs-id1170572548262\">the set of inputs for a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572548267\" class=\"definition\">\n<dt>even function<\/dt>\n<dd id=\"fs-id1170572548272\">a function is even if [latex]f(\u2212x)=f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572548320\" class=\"definition\">\n<dt>function<\/dt>\n<dd id=\"fs-id1170572549920\">a set of inputs, a set of outputs, and a rule for mapping each input to exactly one output<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572549925\" class=\"definition\">\n<dt>graph of a function<\/dt>\n<dd id=\"fs-id1170572549930\">the set of points [latex](x,y)[\/latex] such that [latex]x[\/latex] is in the domain of [latex]f[\/latex] and [latex]y=f(x)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572549980\" class=\"definition\">\n<dt>increasing on the interval [latex]I[\/latex]<\/dt>\n<dd id=\"fs-id1170572549988\">a function increasing on the interval [latex]I[\/latex] if for all [latex]x_1, \\, x_2\\in I, \\, f(x_1)\\le f(x_2)[\/latex] if [latex]x_1<x_2[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572550070\" class=\"definition\">\n<dt>independent variable<\/dt>\n<dd id=\"fs-id1170572550076\">the input variable for a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572550080\" class=\"definition\">\n<dt>odd function<\/dt>\n<dd id=\"fs-id1170572550086\">a function is odd if [latex]f(\u2212x)=\u2212f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572550135\" class=\"definition\">\n<dt>range<\/dt>\n<dd id=\"fs-id1170572235100\">the set of outputs for a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572235105\" class=\"definition\">\n<dt>symmetry about the origin<\/dt>\n<dd id=\"fs-id1170572235110\">the graph of a function [latex]f[\/latex] is symmetric about the origin if [latex](\u2212x,\u2212y)[\/latex] is on the graph of [latex]f[\/latex] whenever [latex](x,y)[\/latex] is on the graph<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572235164\" class=\"definition\">\n<dt>symmetry about the [latex]y[\/latex]-axis<\/dt>\n<dd id=\"fs-id1170572235175\">the graph of a function [latex]f[\/latex] is symmetric about the [latex]y[\/latex]-axis if [latex](\u2212x,y)[\/latex] is on the graph of [latex]f[\/latex] whenever [latex](x,y)[\/latex] is on the graph<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572235230\" class=\"definition\">\n<dt>table of values<\/dt>\n<dd id=\"fs-id1170572235236\">a table containing a list of inputs and their corresponding outputs<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572235240\" class=\"definition\">\n<dt>vertical line test<\/dt>\n<dd id=\"fs-id1170572235245\">given the graph of a function, every vertical line intersects the graph, at most, once<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572235250\" class=\"definition\">\n<dt>zeros of a function<\/dt>\n<dd id=\"fs-id1170572235255\">when a real number [latex]x[\/latex] is a zero of a function [latex]f[\/latex], [latex]f(x)=0[\/latex]<\/dd>\n<\/dl>\n<p>&nbsp;<\/p>\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-102\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/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":17533,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-102","chapter","type-chapter","status-publish","hentry"],"part":21,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/102","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":8,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/102\/revisions"}],"predecessor-version":[{"id":1253,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/102\/revisions\/1253"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/21"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/102\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=102"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=102"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=102"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=102"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}