{"id":2559,"date":"2025-08-13T18:04:17","date_gmt":"2025-08-13T18:04:17","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2559"},"modified":"2026-01-12T19:17:51","modified_gmt":"2026-01-12T19:17:51","slug":"polynomial-functions-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/polynomial-functions-background-youll-need-2\/","title":{"raw":"Polynomial Functions: Background You'll Need 2","rendered":"Polynomial Functions: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Identify even and odd functions<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Determine Whether a Functions is Even, Odd, or Neither<\/h2>\r\nSome functions have symmetry, meaning their graphs remain unchanged when reflected. For example, reflecting the toolkit functions [latex]f(x) = x^2[\/latex] or [latex]f(x) = |x|[\/latex] horizontally across the y-axis will produce the same graph. We call these functions <strong>even functions<\/strong> because they are symmetric about the y-axis.\r\n\r\nIf the graph of [latex]f(x) = x^3[\/latex] or [latex]f(x) = \\dfrac{1}{x}[\/latex] is reflected across both the x-axis and y-axis, the result is also the original graph.\r\n\r\n[caption id=\"\" align=\"alignnone\" 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\" \/> Graph of [latex]x^3[\/latex] and its reflections[\/caption]These graphs are symmetric about the origin, and we call functions with this type of symmetry <strong>odd functions<\/strong>.\r\n\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\u00a0A 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].<\/section><section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>even and odd functions<\/h3>\r\nA function is called an <strong>even function<\/strong> if for every input [latex]x[\/latex]\r\n\r\n&nbsp;\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/p>\r\n&nbsp;\r\n\r\nThe graph of an even function is symmetric about the [latex]y\\text{-}[\/latex] axis.\r\n\r\n&nbsp;\r\n\r\nA function is called an <strong>odd function<\/strong> if for every input [latex]x[\/latex]\r\n\r\n&nbsp;\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex]<\/p>\r\n&nbsp;\r\n\r\nThe graph of an odd function is symmetric about the origin.\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given the formula for a function, determine if the function is even, odd, or neither.\r\n<\/strong>\r\n<ol>\r\n \t<li>Determine whether the function satisfies [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex]. If it does, it is even.<\/li>\r\n \t<li>Determine whether the function satisfies [latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex]. If it does, it is odd.<\/li>\r\n \t<li>If the function does not satisfy either rule, it is neither even nor odd.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Is the function [latex]f\\left(x\\right)={x}^{3}+2x[\/latex] even, odd, or neither?[reveal-answer q=\"936347\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"936347\"]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.\r\n<p style=\"text-align: center;\">[latex]f\\left(-x\\right)={\\left(-x\\right)}^{3}+2\\left(-x\\right)=-{x}^{3}-2x[\/latex]<\/p>\r\nThis does not return us to the original function, so this function is not even. We can now test the rule for odd functions.\r\n<p style=\"text-align: center;\">[latex]-f\\left(-x\\right)=-\\left(-{x}^{3}-2x\\right)={x}^{3}+2x[\/latex]<\/p>\r\nBecause [latex]-f\\left(-x\\right)=f\\left(x\\right)[\/latex], this is an odd function.\r\n\r\n<strong>Analysis of the Solution<\/strong>\r\n\r\nConsider 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.\r\n\r\n<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\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]318779[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Identify even and odd functions<\/li>\n<\/ul>\n<\/section>\n<h2>Determine Whether a Functions is Even, Odd, or Neither<\/h2>\n<p>Some functions have symmetry, meaning their graphs remain unchanged when reflected. For example, reflecting the toolkit functions [latex]f(x) = x^2[\/latex] or [latex]f(x) = |x|[\/latex] horizontally across the y-axis will produce the same graph. We call these functions <strong>even functions<\/strong> because they are symmetric about the y-axis.<\/p>\n<p>If the graph of [latex]f(x) = x^3[\/latex] or [latex]f(x) = \\dfrac{1}{x}[\/latex] is reflected across both the x-axis and y-axis, the result is also the original graph.<\/p>\n<figure style=\"width: 975px\" class=\"wp-caption alignnone\"><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\">Graph of [latex]x^3[\/latex] and its reflections<\/figcaption><\/figure>\n<p>These graphs are symmetric about the origin, and we call functions with this type of symmetry <strong>odd functions<\/strong>.<\/p>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\u00a0A 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].<\/section>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\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 questionHelp\" aria-label=\"Question Help\"><strong>How To: Given the formula for a function, determine if the function is even, odd, or neither.<br \/>\n<\/strong><\/p>\n<ol>\n<li>Determine whether the function satisfies [latex]f\\left(x\\right)=f\\left(-x\\right)[\/latex]. If it does, it is even.<\/li>\n<li>Determine whether the function satisfies [latex]f\\left(x\\right)=-f\\left(-x\\right)[\/latex]. If it does, it is odd.<\/li>\n<li>If the function does not satisfy either rule, it is neither even nor odd.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Is the function [latex]f\\left(x\\right)={x}^{3}+2x[\/latex] even, odd, or neither?<\/p>\n<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\">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><strong>Analysis of the Solution<\/strong><\/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 tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm318779\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318779&theme=lumen&iframe_resize_id=ohm318779&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":67,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":74,"module-header":"background_you_need","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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2559"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2559\/revisions"}],"predecessor-version":[{"id":5291,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2559\/revisions\/5291"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/74"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2559\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2559"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2559"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2559"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2559"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}