{"id":375,"date":"2025-02-13T19:44:25","date_gmt":"2025-02-13T19:44:25","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/introduction-to-integration-background-youll-need-3\/"},"modified":"2025-02-13T19:44:25","modified_gmt":"2025-02-13T19:44:25","slug":"introduction-to-integration-background-youll-need-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/introduction-to-integration-background-youll-need-3\/","title":{"raw":"Introduction to Integration: Background You'll Need 3","rendered":"Introduction to Integration: Background You&#8217;ll Need 3"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Determine whether a function is even, odd, or neither&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6915,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:{&quot;1&quot;:2,&quot;2&quot;:16777215},&quot;11&quot;:4,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Determine whether a function is even, odd, or neither<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Determine Whether a Functions is Even, Odd, or Neither<\/h2>\n<p>Functions often display specific symmetries that define their characteristics. For instance:<\/p>\n<ul>\n\t<li><strong>Even Functions<\/strong>\n<ul>\n\t<li>A function [latex]f(x)[\/latex] is called even if it is symmetric about the [latex]y[\/latex]-axis. This means [latex]f(\u2212x)=f(x)[\/latex] for all [latex]x[\/latex].<\/li>\n\t<li>Graphically, this symmetry means that if the graph of the function is folded along the [latex]y[\/latex]-axis, the two halves will match exactly.<\/li>\n\t<li>Examples include [latex]f(x)=x^2[\/latex] or [latex]f(x)=\u2223x\u2223[\/latex], where horizontal reflections produce the original graph.<\/li>\n<\/ul>\n<\/li>\n\t<li><strong>Odd Functions<\/strong>\n<ul>\n\t<li>A function [latex]f(x)[\/latex] is called odd if it has rotational symmetry about the origin, which means [latex]f(\u2212x)=\u2212f(x)[\/latex] for all [latex]x[\/latex].<\/li>\n\t<li>This property implies that if the function's graph is rotated [latex]180[\/latex] degrees about the origin, it will coincide with its original shape.<\/li>\n\t<li>An example is [latex]f(x)=x^3[\/latex], where reflecting the graph both horizontally and vertically reproduces the original graph.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<section class=\"textbox example\">\n<p>The function [latex]f(x)=x^3[\/latex] demonstrates odd symmetry. As shown in the graphs below:<\/p>\n\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\/18203605\/CNX_Precalc_Figure_01_05_021abc2.jpg\" alt=\"Graph of x^3 and its reflections.\" width=\"975\" height=\"407\"> (a) The cubic toolkit function (b) Horizontal reflection of the cubic toolkit function (c) Horizontal and vertical reflections reproduce the original cubic function.<span style=\"font-size: 16px;\">&nbsp;<\/span>[\/caption]\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>even and odd functions<\/h3>\n<p>A function is called an <strong>even function<\/strong> if for every input [latex]x[\/latex],<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The graph of an even function is symmetric about the [latex]y\\text{-}[\/latex] axis.<\/p>\n<p>&nbsp;<\/p>\n<p>A function is called an <strong>odd function<\/strong> if for every input [latex]x[\/latex],<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The graph of an odd function is symmetric about the origin.<\/p>\n<\/section>\n<section class=\"textbox proTip\">\n<p>A function can be neither even nor odd if it does not exhibit either symmetry. For example, [latex]f\\left(x\\right)={2}^{x}[\/latex] is neither even nor odd. Also, the only function that is both even and odd is the constant function [latex]f\\left(x\\right)=0[\/latex].<\/p>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Determine If a Function is Even, Odd, or Neither<\/strong><\/p>\n<ul>\n\t<li><strong>Check for Even Symmetry<\/strong>:\n\n<ul>\n\t<li>Evaluate [latex]f(\u2212x)[\/latex] and compare it with [latex]f(x)[\/latex]<\/li>\n\t<li>If [latex]f(\u2212x)=f(x)[\/latex] for all values of [latex]x[\/latex] in the domain of the function, then the function is <strong>even<\/strong>.<\/li>\n<\/ul>\n<\/li>\n\t<li><strong>Check for Odd Symmetry<\/strong>:\n\n<ul>\n\t<li>Evaluate [latex]f(\u2212x)[\/latex] and compare it with [latex]f(x)[\/latex]<\/li>\n\t<li>If [latex]f(\u2212x)=\u2212f(x)[\/latex] for all values of [latex]x[\/latex], then the function is <strong>odd<\/strong>.<\/li>\n<\/ul>\n<\/li>\n\t<li><strong>Neither Even nor Odd<\/strong>: If neither of the above conditions is met, the function is neither even nor odd.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">\n<p>Is the function [latex]f\\left(x\\right)={x}^{3}+2x[\/latex] even, odd, or neither?<\/p>\n<p>[reveal-answer q=\"936347\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"936347\"]<\/p>\n<p>Without looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections and determining if they return us to the original function. Let\u2019s begin with the rule for even functions.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(-x\\right)={\\left(-x\\right)}^{3}+2\\left(-x\\right)=-{x}^{3}-2x[\/latex]<\/p>\n<p>This does not return us to the original function, so this function is not even. We can now test the rule for odd functions.<\/p>\n<p style=\"text-align: center;\">[latex]-f\\left(-x\\right)=-\\left(-{x}^{3}-2x\\right)={x}^{3}+2x[\/latex]<\/p>\n<p>Because [latex]-f\\left(-x\\right)=f\\left(x\\right)[\/latex], this is an odd function.<\/p>\n<p>Consider the graph of [latex]f[\/latex]. Notice that the graph is symmetric about the origin. For every point [latex]\\left(x,y\\right)[\/latex] on the graph, the corresponding point [latex]\\left(-x,-y\\right)[\/latex] is also on the graph. For example, (1, 3) is on the graph of [latex]f[\/latex], and the corresponding point [latex]\\left(-1,-3\\right)[\/latex] is also on the graph.<\/p>\n<p><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18203607\/CNX_Precalc_Figure_01_05_0392.jpg\" alt=\"Graph of f(x) with labeled points at (1, 3) and (-1, -3).\" width=\"731\" height=\"488\"><\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Is the function [latex]f\\left(s\\right)={s}^{4}+3{s}^{2}+7[\/latex] even, odd, or neither?<\/p>\n<p>[reveal-answer q=\"630369\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"630369\"]<\/p>\n<p>Even<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox tryIt\">\n<p>[ohm_question hide_question_numbers=1]293883[\/ohm_question]<\/p>\n<\/section>\n<section class=\"textbox watchIt\">\n<p><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=6454976&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=VvUI6E78cN4&amp;video_target=tpm-plugin-saqsqzpb-VvUI6E78cN4\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/IntroductionToOddAndEvenFunctions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \"Introduction to Odd and Even Functions\" here (opens in new window)<\/a>.<\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Determine whether a function is even, odd, or neither&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6915,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:{&quot;1&quot;:2,&quot;2&quot;:16777215},&quot;11&quot;:4,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:0},&quot;15&quot;:&quot;Calibri&quot;}\">Determine whether a function is even, odd, or neither<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Determine Whether a Functions is Even, Odd, or Neither<\/h2>\n<p>Functions often display specific symmetries that define their characteristics. For instance:<\/p>\n<ul>\n<li><strong>Even Functions<\/strong>\n<ul>\n<li>A function [latex]f(x)[\/latex] is called even if it is symmetric about the [latex]y[\/latex]-axis. This means [latex]f(\u2212x)=f(x)[\/latex] for all [latex]x[\/latex].<\/li>\n<li>Graphically, this symmetry means that if the graph of the function is folded along the [latex]y[\/latex]-axis, the two halves will match exactly.<\/li>\n<li>Examples include [latex]f(x)=x^2[\/latex] or [latex]f(x)=\u2223x\u2223[\/latex], where horizontal reflections produce the original graph.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Odd Functions<\/strong>\n<ul>\n<li>A function [latex]f(x)[\/latex] is called odd if it has rotational symmetry about the origin, which means [latex]f(\u2212x)=\u2212f(x)[\/latex] for all [latex]x[\/latex].<\/li>\n<li>This property implies that if the function&#8217;s graph is rotated [latex]180[\/latex] degrees about the origin, it will coincide with its original shape.<\/li>\n<li>An example is [latex]f(x)=x^3[\/latex], where reflecting the graph both horizontally and vertically reproduces the original graph.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<section class=\"textbox example\">\n<p>The function [latex]f(x)=x^3[\/latex] demonstrates odd symmetry. As shown in the graphs below:<\/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\/18203605\/CNX_Precalc_Figure_01_05_021abc2.jpg\" alt=\"Graph of x^3 and its reflections.\" width=\"975\" height=\"407\" \/><figcaption class=\"wp-caption-text\">(a) The cubic toolkit function (b) Horizontal reflection of the cubic toolkit function (c) Horizontal and vertical reflections reproduce the original cubic function.<span style=\"font-size: 16px;\">&nbsp;<\/span><\/figcaption><\/figure>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>even and odd functions<\/h3>\n<p>A function is called an <strong>even function<\/strong> if for every input [latex]x[\/latex],<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The graph of an even function is symmetric about the [latex]y\\text{-}[\/latex] axis.<\/p>\n<p>&nbsp;<\/p>\n<p>A function is called an <strong>odd function<\/strong> if for every input [latex]x[\/latex],<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The graph of an odd function is symmetric about the origin.<\/p>\n<\/section>\n<section class=\"textbox proTip\">\n<p>A function can be neither even nor odd if it does not exhibit either symmetry. For example, [latex]f\\left(x\\right)={2}^{x}[\/latex] is neither even nor odd. Also, the only function that is both even and odd is the constant function [latex]f\\left(x\\right)=0[\/latex].<\/p>\n<\/section>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Determine If a Function is Even, Odd, or Neither<\/strong><\/p>\n<ul>\n<li><strong>Check for Even Symmetry<\/strong>:\n<ul>\n<li>Evaluate [latex]f(\u2212x)[\/latex] and compare it with [latex]f(x)[\/latex]<\/li>\n<li>If [latex]f(\u2212x)=f(x)[\/latex] for all values of [latex]x[\/latex] in the domain of the function, then the function is <strong>even<\/strong>.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Check for Odd Symmetry<\/strong>:\n<ul>\n<li>Evaluate [latex]f(\u2212x)[\/latex] and compare it with [latex]f(x)[\/latex]<\/li>\n<li>If [latex]f(\u2212x)=\u2212f(x)[\/latex] for all values of [latex]x[\/latex], then the function is <strong>odd<\/strong>.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Neither Even nor Odd<\/strong>: If neither of the above conditions is met, the function is neither even nor odd.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">\n<p>Is the function [latex]f\\left(x\\right)={x}^{3}+2x[\/latex] even, odd, or neither?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q936347\">Show Solution<\/button><\/p>\n<div id=\"q936347\" class=\"hidden-answer\" style=\"display: none\">\n<p>Without looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections and determining if they return us to the original function. Let\u2019s begin with the rule for even functions.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(-x\\right)={\\left(-x\\right)}^{3}+2\\left(-x\\right)=-{x}^{3}-2x[\/latex]<\/p>\n<p>This does not return us to the original function, so this function is not even. We can now test the rule for odd functions.<\/p>\n<p style=\"text-align: center;\">[latex]-f\\left(-x\\right)=-\\left(-{x}^{3}-2x\\right)={x}^{3}+2x[\/latex]<\/p>\n<p>Because [latex]-f\\left(-x\\right)=f\\left(x\\right)[\/latex], this is an odd function.<\/p>\n<p>Consider the graph of [latex]f[\/latex]. Notice that the graph is symmetric about the origin. For every point [latex]\\left(x,y\\right)[\/latex] on the graph, the corresponding point [latex]\\left(-x,-y\\right)[\/latex] is also on the graph. For example, (1, 3) is on the graph of [latex]f[\/latex], and the corresponding point [latex]\\left(-1,-3\\right)[\/latex] is also on the graph.<\/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\/18203607\/CNX_Precalc_Figure_01_05_0392.jpg\" alt=\"Graph of f(x) with labeled points at (1, 3) and (-1, -3).\" width=\"731\" height=\"488\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Is the function [latex]f\\left(s\\right)={s}^{4}+3{s}^{2}+7[\/latex] even, odd, or neither?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q630369\">Show Solution<\/button><\/p>\n<div id=\"q630369\" class=\"hidden-answer\" style=\"display: none\">\n<p>Even<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm293883\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=293883&theme=lumen&iframe_resize_id=ohm293883&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=6454976&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=VvUI6E78cN4&amp;video_target=tpm-plugin-saqsqzpb-VvUI6E78cN4\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/IntroductionToOddAndEvenFunctions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for &#8220;Introduction to Odd and Even Functions&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n","protected":false},"author":6,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":371,"module-header":"","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/375"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/375\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/371"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/375\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=375"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=375"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=375"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=375"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}