{"id":7956,"date":"2023-05-06T21:50:00","date_gmt":"2023-05-06T16:20:00","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=7956"},"modified":"2023-05-02T04:35:26","modified_gmt":"2023-05-01T23:05:26","slug":"even-functions-definition-graph-examples","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/even-functions-definition-graph-examples\/","title":{"rendered":"Even Functions : Definition , Graph ,examples"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">What is even functions<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>An even function is a function f(x) that satisfies the property f(x) = f(-x) for all x in the domain of the function. <\/li>\n\n\n\n<li>In other words, an even function is symmetric with respect to the y-axis. <\/li>\n\n\n\n<li>Geometrically, an even function has the property that if you fold the graph of the function along the y-axis, the two halves of the graph overlap perfectly.<\/li>\n\n\n\n<li>Graph of even function is symmetrical in I and II  or III and IV quadrants<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Examples<\/h2>\n\n\n\n<p>(1) <strong>Polynomial functions<\/strong><\/p>\n\n\n\n<p>Any&nbsp;<a href=\"https:\/\/physicscatalyst.com\/Class9\/polynomial.php\">polynomial<\/a>&nbsp;with only even degree terms is an even function. For example, f(x) = 6x<sup>8<\/sup>&nbsp;&#8211; 6x<sup>2<\/sup>&nbsp;&#8211; 5.<\/p>\n\n\n\n<p>f(-x)= 6(-x)^8 -6(-x)^2 &#8211; 5 =6x^8 -6x^2 -5= f(x)<\/p>\n\n\n\n<p>(2)  Co<strong>sine Function<\/strong> [cos(x)]<\/p>\n\n\n\n<p>The cosine function is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. It is an even function because cos(x) = cos(-x) for all x. This property can also be derived from the geometric interpretation of cosine, which is that it represents the x-coordinate of a point on the unit circle.<\/p>\n\n\n\n<p>(3)<strong> Parabola Function<\/strong>: $x^2$<\/p>\n\n\n\n<p>The parabola function is defined as f(x) = x^2. It is an even function because f(x) = f(-x) for all x. This property can be seen by observing that the graph of the function is symmetric about the y-axis<\/p>\n\n\n\n<p>(4)  <strong>Absolute Value Function<\/strong>: |x|<\/p>\n\n\n\n<p>The absolute value function is defined as f(x) = |x|. It is an even function because f(x) = f(-x) for all x. This can be seen by using the definition of absolute value, which is that |x| is equal to x when x is positive and -x when x is negative.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Properties of even function<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For any function f(x), f(x) + f(-x) is an even function.<\/li>\n\n\n\n<li>The sum or difference of two even functions is even.<\/li>\n<\/ul>\n\n\n\n<p>let f(x)= f(-x) and g(x) = g(-x)<\/p>\n\n\n\n<p>Now h(x) = f(x) + g(x)<br>h(-x) = f(-x) +g(-x) =f(x) + g(x)= h(x)<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The multiple of an even function is again an even function.<\/li>\n<\/ul>\n\n\n\n<p>let f(x)= f(-x)  and h(x) = n f(x)<br>then<br>h(-x) = nf(-x) = nf(x) = h(x)<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The product or division of two even functions is even.<\/li>\n<\/ul>\n\n\n\n<p>let f(x)= f(-x) and g(x) = g(-x)<\/p>\n\n\n\n<p>Now h(x) = f(x)  g(x)<br>h(-x) = f(-x) g(-x) =f(x)  g(x)= h(x)<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The composition of two even functions and the composition of an even and odd function is even.<\/li>\n<\/ul>\n\n\n\n<p>let Both even f(x)= f(-x) and g(x) = g(-x)<\/p>\n\n\n\n<p>$f \\circ g (x) = f(g(x)$<br>Now $f \\circ g (-x)= f(g(-x)= f(g(x)$<\/p>\n\n\n\n<p> let one even f(x)= f(-x) and g(x) =- g(-x)<\/p>\n\n\n\n<p>$f \\circ g (x) = f(g(x))$<br>Now $f \\circ g (-x)= f(g(-x)= f(-g(x))=f(g(x))$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">How to Prove the even Function<\/h2>\n\n\n\n<p>We simply need to prove that f(x) = f(-x)<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Solved Examples<\/h2>\n\n\n\n<p><strong>Example 1<\/strong><\/p>\n\n\n\n<p>$f(x) = \\cos (sin^3 x)$<\/p>\n\n\n\n<p><strong>Solution<\/strong><\/p>\n\n\n\n<p>$f(-x)= \\cos (sin^3 -x)= \\cos [(-sin x)^3]= \\cos (- \\sin^3 x) = \\cos  (sin^3 x) = f(x)$<\/p>\n\n\n\n<p>So it is even function<\/p>\n\n\n\n<p><strong>Example 2<\/strong><\/p>\n\n\n\n<p>f(x) = x<sup>2<\/sup>&nbsp;+ 3x \u2013 6<\/p>\n\n\n\n<p><strong>Solution<\/strong><\/p>\n\n\n\n<p>$f(-x) = (-x)^2 + 3(-x) -6 = x^2 -3x-6 \\neq f(x)$<br>hence this is not even function<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>What is even functions Examples (1) Polynomial functions Any&nbsp;polynomial&nbsp;with only even degree terms is an even function. For example, f(x) = 6&#215;8&nbsp;&#8211; 6&#215;2&nbsp;&#8211; 5. f(-x)= 6(-x)^8 -6(-x)^2 &#8211; 5 =6x^8 -6x^2 -5= f(x) (2) Cosine Function [cos(x)] The cosine function is defined as the ratio of the adjacent side to the hypotenuse in a right-angled [&hellip;]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[498],"tags":[],"class_list":["post-7956","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Even Functions : Definition , Graph ,examples - physicscatalyst&#039;s Blog<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/physicscatalyst.com\/article\/even-functions-definition-graph-examples\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Even Functions : Definition , Graph ,examples - physicscatalyst&#039;s Blog\" \/>\n<meta property=\"og:description\" content=\"What is even functions Examples (1) Polynomial functions Any&nbsp;polynomial&nbsp;with only even degree terms is an even function. 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For example, f(x) = 6x8&nbsp;&#8211; 6x2&nbsp;&#8211; 5. f(-x)= 6(-x)^8 -6(-x)^2 &#8211; 5 =6x^8 -6x^2 -5= f(x) (2) Cosine Function [cos(x)] The cosine function is defined as the ratio of the adjacent side to the hypotenuse in a right-angled&hellip;","_links":{"self":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/7956","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/comments?post=7956"}],"version-history":[{"count":2,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/7956\/revisions"}],"predecessor-version":[{"id":7961,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/7956\/revisions\/7961"}],"wp:attachment":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/media?parent=7956"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/categories?post=7956"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/tags?post=7956"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}