{"id":43263,"date":"2025-06-29T17:00:20","date_gmt":"2025-06-29T17:00:20","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=43263"},"modified":"2025-06-29T17:00:22","modified_gmt":"2025-06-29T17:00:22","slug":"determine-algebraically-whether-the-function-is-even-odd-or-neither","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/determine-algebraically-whether-the-function-is-even-odd-or-neither\/","title":{"rendered":"Determine algebraically whether the function is even, odd, or neither"},"content":{"rendered":"\n<p>Determine algebraically whether the function is even, odd, or neither. 11) f(x) = 8x^3 + 3 12) f(x) = -2x^2 + 4 13) f(x) = 4x^3<\/p>\n\n\n\n<p><strong>The Correct Answer and Explanation is:<\/strong><\/p>\n\n\n\n<p>To determine whether a function is even, odd, or neither, we use the following definitions:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Even function:<\/strong> A function is even if for all xxx, f(\u2212x)=f(x)f(-x) = f(x)f(\u2212x)=f(x).<\/li>\n\n\n\n<li><strong>Odd function:<\/strong> A function is odd if for all xxx, f(\u2212x)=\u2212f(x)f(-x) = -f(x)f(\u2212x)=\u2212f(x).<\/li>\n\n\n\n<li><strong>Neither:<\/strong> If the function doesn&#8217;t satisfy the conditions for being even or odd, then it is neither.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">11) f(x)=8&#215;3+3f(x) = 8x^3 + 3f(x)=8&#215;3+3<\/h3>\n\n\n\n<p>To check whether this function is even or odd, we need to compute f(\u2212x)f(-x)f(\u2212x) and compare it to f(x)f(x)f(x) and \u2212f(x)-f(x)\u2212f(x).<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>First, calculate f(\u2212x)f(-x)f(\u2212x): f(\u2212x)=8(\u2212x)3+3=\u22128&#215;3+3f(-x) = 8(-x)^3 + 3 = -8x^3 + 3f(\u2212x)=8(\u2212x)3+3=\u22128&#215;3+3<\/li>\n\n\n\n<li>Now, compare f(\u2212x)f(-x)f(\u2212x) to f(x)f(x)f(x) and \u2212f(x)-f(x)\u2212f(x):\n<ul class=\"wp-block-list\">\n<li>f(x)=8&#215;3+3f(x) = 8x^3 + 3f(x)=8&#215;3+3<\/li>\n\n\n\n<li>\u2212f(x)=\u2212(8&#215;3+3)=\u22128&#215;3\u22123-f(x) = -(8x^3 + 3) = -8x^3 &#8211; 3\u2212f(x)=\u2212(8&#215;3+3)=\u22128&#215;3\u22123<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p>Since f(\u2212x)=\u22128&#215;3+3f(-x) = -8x^3 + 3f(\u2212x)=\u22128&#215;3+3 is not equal to f(x)f(x)f(x) nor to \u2212f(x)-f(x)\u2212f(x), the function is <strong>neither even nor odd<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">12) f(x)=\u22122&#215;2+4f(x) = -2x^2 + 4f(x)=\u22122&#215;2+4<\/h3>\n\n\n\n<p>Again, we compute f(\u2212x)f(-x)f(\u2212x) and compare it to f(x)f(x)f(x) and \u2212f(x)-f(x)\u2212f(x).<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>First, calculate f(\u2212x)f(-x)f(\u2212x): f(\u2212x)=\u22122(\u2212x)2+4=\u22122&#215;2+4f(-x) = -2(-x)^2 + 4 = -2x^2 + 4f(\u2212x)=\u22122(\u2212x)2+4=\u22122&#215;2+4<\/li>\n\n\n\n<li>Now, compare f(\u2212x)f(-x)f(\u2212x) to f(x)f(x)f(x) and \u2212f(x)-f(x)\u2212f(x):\n<ul class=\"wp-block-list\">\n<li>f(x)=\u22122&#215;2+4f(x) = -2x^2 + 4f(x)=\u22122&#215;2+4<\/li>\n\n\n\n<li>\u2212f(x)=\u2212(\u22122&#215;2+4)=2&#215;2\u22124-f(x) = -(-2x^2 + 4) = 2x^2 &#8211; 4\u2212f(x)=\u2212(\u22122&#215;2+4)=2&#215;2\u22124<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p>Since f(\u2212x)=f(x)f(-x) = f(x)f(\u2212x)=f(x), the function is <strong>even<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">13) f(x)=4x3f(x) = 4x^3f(x)=4&#215;3<\/h3>\n\n\n\n<p>Once again, we compute f(\u2212x)f(-x)f(\u2212x) and compare it to f(x)f(x)f(x) and \u2212f(x)-f(x)\u2212f(x).<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>First, calculate f(\u2212x)f(-x)f(\u2212x): f(\u2212x)=4(\u2212x)3=\u22124x3f(-x) = 4(-x)^3 = -4x^3f(\u2212x)=4(\u2212x)3=\u22124&#215;3<\/li>\n\n\n\n<li>Now, compare f(\u2212x)f(-x)f(\u2212x) to f(x)f(x)f(x) and \u2212f(x)-f(x)\u2212f(x):\n<ul class=\"wp-block-list\">\n<li>f(x)=4x3f(x) = 4x^3f(x)=4&#215;3<\/li>\n\n\n\n<li>\u2212f(x)=\u2212(4&#215;3)=\u22124&#215;3-f(x) = -(4x^3) = -4x^3\u2212f(x)=\u2212(4&#215;3)=\u22124&#215;3<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p>Since f(\u2212x)=\u2212f(x)f(-x) = -f(x)f(\u2212x)=\u2212f(x), the function is <strong>odd<\/strong>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Summary of Results:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>f(x) = 8x^3 + 3<\/strong> is <strong>neither<\/strong> even nor odd.<\/li>\n\n\n\n<li><strong>f(x) = -2x^2 + 4<\/strong> is <strong>even<\/strong>.<\/li>\n\n\n\n<li><strong>f(x) = 4x^3<\/strong> is <strong>odd<\/strong>.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"852\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1457.jpeg\" alt=\"\" class=\"wp-image-43264\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1457.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1457-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1457-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Determine algebraically whether the function is even, odd, or neither. 11) f(x) = 8x^3 + 3 12) f(x) = -2x^2 + 4 13) f(x) = 4x^3 The Correct Answer and Explanation is: To determine whether a function is even, odd, or neither, we use the following definitions: 11) f(x)=8&#215;3+3f(x) = 8x^3 + 3f(x)=8&#215;3+3 To check [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-43263","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/43263","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=43263"}],"version-history":[{"count":2,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/43263\/revisions"}],"predecessor-version":[{"id":43266,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/43263\/revisions\/43266"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=43263"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=43263"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=43263"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}