{"id":86581,"date":"2021-07-23T11:22:06","date_gmt":"2021-07-23T16:22:06","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=86581"},"modified":"2026-03-28T10:28:48","modified_gmt":"2026-03-28T15:28:48","slug":"determining-even-and-odd-functions","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/determining-even-and-odd-functions\/","title":{"rendered":"Even and Odd Functions"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_uFTCkspiO3o\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_uFTCkspiO3o\" data-source-videoID=\"uFTCkspiO3o\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Even and Odd Functions Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Even and Odd Functions\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_uFTCkspiO3o:hover {cursor:pointer;} img#videoThumbnailImage_uFTCkspiO3o {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/1720-thumb-final-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_uFTCkspiO3o\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_uFTCkspiO3o\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Even and Odd Functions\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_uFTCkspiO3o\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_uFTCkspiO3o\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_uFTCkspiO3o\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction 6dw_Function() {\n  var x = document.getElementById(\"6dw\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"6dw_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"6dw\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Even_Functions\" class=\"smooth-scroll\">Even Functions<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Odd_Functions\" class=\"smooth-scroll\">Odd Functions<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Functions_That_Are_Neither_Even_Nor_Odd\" class=\"smooth-scroll\">Functions That Are Neither Even Nor Odd<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#The_Reason_for_Even_and_Odd_Functions\" class=\"smooth-scroll\">The Reason for Even and Odd Functions<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Review_Questions\" class=\"smooth-scroll\">Review Questions<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Even_and_Odd_Function_Practice_Questions\" class=\"smooth-scroll\">Even and Odd Function Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>When we think \u201ceven and odd,\u201d usually even and odd numbers are what come to mind. But what are even and odd functions? In today\u2019s video, we will define even and odd functions and discuss how to identify them.<\/p>\n<h2><span id=\"Even_Functions\" class=\"m-toc-anchor\"><\/span>Even Functions<\/h2>\n<p>\nLet\u2019s begin by talking about even functions. If a function <span style=\"font-style:normal; font-size:90%\">\\(f(x)\\)<\/span> evaluated at <span style=\"font-style:normal; font-size:90%\">\\(-x\\)<\/span> gives us the same <span style=\"font-style:normal; font-size:90%\">\\(f(x)\\)<\/span> we started with, that function is even.<\/p>\n<p>Formally written:<\/p>\n<div class=\"examplesentence\">\\(f(x)=f(-x)\\)<\/div>\n<p>\n&nbsp;<br \/>\n<span style=\"font-style:normal; font-size:90%\">\\(f(x)\\)<\/span> is <strong>even<\/strong>. Graphically, an even function will appear symmetric, or mirrored, about the vertical y-axis. One example of an even function is <span style=\"font-style:normal; font-size:90%\">\\(f(x)=x^{2}\\)<\/span>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-89935\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Even-and-Odd-functions-1.png\" alt=\"f(x)=x^2 on a graph. The sides of the curve are identical. \" width=\"777\" height=\"437\"style=\"box-shadow: 1.5px 1.5px 3px grey;\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Even-and-Odd-functions-1.png 1920w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Even-and-Odd-functions-1-300x168.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Even-and-Odd-functions-1-1024x574.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Even-and-Odd-functions-1-768x431.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Even-and-Odd-functions-1-1536x862.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>Notice that the shape of this familiar parabola is visibly symmetric. The left and right sides of the plane are identical, just flipped. We can also show that this function is even algebraically, by evaluating at <span style=\"font-style:normal; font-size:90%\">\\(-x\\)<\/span>. <\/p>\n<p>So our original function is <span style=\"font-style:normal; font-size:90%\">\\(f(x)=x^{2}\\)<\/span>. And we said that, if <span style=\"font-style:normal; font-size:90%\">\\(f(-x)\\)<\/span> is the same as <span style=\"font-style:normal; font-size:90%\">\\(f(x)\\)<\/span>, then the function is even. So let\u2019s evaluate at <span style=\"font-style:normal; font-size:90%\">\\(-x\\)<\/span>. So, wherever there\u2019s an <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>, we\u2019re gonna plug in <span style=\"font-style:normal; font-size:90%\">\\(-x\\)<\/span>. <\/p>\n<p>So we have:<\/p>\n<div class=\"examplesentence\">\\(f(-x)=(-x)^{2}\\)<\/div>\n<p>\n&nbsp;<br \/>\nWhich, when you square a negative, it turns positive, so this is equal to:<\/p>\n<div class=\"examplesentence\">\\(f(-x)=(-x)^{2}=x^{2}\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo in this case, <span style=\"font-style:normal; font-size:90%\">\\(f(-x)=f(x)\\)<\/span>. And because of our definition of even, this function, <span style=\"font-style:normal; font-size:90%\">\\(f(x)=x^{2}\\)<\/span>, is even.<\/p>\n<p>Notice that if we add a constant to this function, it won\u2019t affect the shape of the function, just raise or lower it on the plane. For example, this is the graph of <span style=\"font-style:normal; font-size:90%\">\\(f(x)=x^{2}+1\\)<\/span>. Notice how it still has the same shape, and it is still an even function. It has just been moved up one unit on the coordinate plane.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Even-and-Odd-Functions-2.png\" alt=\"Image of f(x)=x2+1 written on a graph. The point is raised. \" width=\"777\" height=\"437\" class=\"aligncenter size-full wp-image-89941\"style=\"box-shadow: 1.5px 1.5px 3px grey\" \/><\/p>\n<h2><span id=\"Odd_Functions\" class=\"m-toc-anchor\"><\/span>Odd Functions<\/h2>\n<p>\nNow let\u2019s talk about what odd functions are like.<\/p>\n<p>Consider another function <span style=\"font-style:normal; font-size:90%\">\\(f(x)\\)<\/span>, which we will once again evaluate at <span style=\"font-style:normal; font-size:90%\">\\(-x\\)<\/span>. But this time, instead of looking for the same <span style=\"font-style:normal; font-size:90%\">\\(f(x)\\)<\/span> we started with, we want to see if <span style=\"font-style:normal; font-size:90%\">\\(f(-x)\\)<\/span> changes the sign of all terms in the function.<\/p>\n<p>In other words, if <span style=\"font-style:normal; font-size:90%\">\\(f(-x)=-f(x)\\)<\/span>, then the function is <strong>odd<\/strong>. <\/p>\n<p>Graphically, an odd function will appear the same when we rotate it by 180\u00b0, like flipping a page upside down, and it must pass through the origin. <\/p>\n<p>A few examples of odd functions are:<\/p>\n<div class=\"examplesentence\">\\(f(x)=x\\)<br \/>\n\\(f(x)=x^{3}\\)<br \/>\n\\(f(x)=sin(x)\\)<\/div>\n<p>\n&nbsp;<br \/>\nLet\u2019s take a look at what\u2019s going on here algebraically, using <span style=\"font-style:normal; font-size:90%\">\\(f(x)=x^{3}\\)<\/span> as an example.<\/p>\n<p>So we\u2019re gonna have our original function: <span style=\"font-style:normal; font-size:90%\">\\(f(x)=x^{3}\\)<\/span>. And just like before, we\u2019re gonna evaluate it at <span style=\"font-style:normal; font-size:90%\">\\(-x\\)<\/span>. So we want <span style=\"font-style:normal; font-size:90%\">\\(f(-x)\\)<\/span>. So anywhere we see an <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>, we\u2019re gonna plug in <span style=\"font-style:normal; font-size:90%\">\\(-x\\)<\/span>.<\/p>\n<div class=\"examplesentence\">\\(f(-x)=(-x)^{3}\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo <span style=\"font-style:normal; font-size:90%\">\\((-x)^{3}\\)<\/span>, which is <span style=\"font-style:normal; font-size:90%\">\\(-x\\cdot-x\\cdot-x\\)<\/span>, which means, since there are three negatives, our final answer\u2019s gonna be <span style=\"font-style:normal; font-size:90%\">\\(-x^{3}\\)<\/span>.<\/p>\n<div class=\"examplesentence\">\\(f(-x)=-x^{3}\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo if you notice, <span style=\"font-style:normal; font-size:90%\">\\(f(x)\\)<\/span> is the opposite of <span style=\"font-style:normal; font-size:90%\">\\(f(-x)\\)<\/span>. Each term, which in this case we only have one, is changed from positive to negative. So that means that this function is odd.<\/p>\n<p>Notice that if we were to add a constant to this function, it would no longer be odd. Remember that for odd functions, every term must change signs when evaluating at <span style=\"font-style:normal; font-size:90%\">\\(-x\\)<\/span>. That constant term would have no way to change sign, and we would see on the graph that the function would no longer pass through the origin.<\/p>\n<h2><span id=\"Functions_That_Are_Neither_Even_Nor_Odd\" class=\"m-toc-anchor\"><\/span>Functions That Are Neither Even Nor Odd<\/h2>\n<p>\nSo we have now talked about definitions of both even and odd functions but before we go further, it\u2019s important to clarify that some functions may be neither even nor odd! For example, take a look at the function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=(x+1)^{2}\\)<\/span>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-89944\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Even-and-Odd-Functions-3.png\" alt=\"Image demonstrating that some graphs have equations that are neither even nor odd. \" width=\"777\" height=\"437\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Even-and-Odd-Functions-3.png 1920w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Even-and-Odd-Functions-3-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Even-and-Odd-Functions-3-1024x576.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Even-and-Odd-Functions-3-768x432.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Even-and-Odd-Functions-3-1536x864.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>From the graph we can see that this function doesn\u2019t pass through the origin, so it can\u2019t be odd. And it isn\u2019t symmetric about the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis, so it isn\u2019t even either. But we can also determine this algebraically.<\/p>\n<p>So our function is <span style=\"font-style:normal; font-size:90%\">\\(f(x)=(x+1)^{2}\\)<\/span>. And remember, to determine if it\u2019s even or odd, we want to evaluate it at <span style=\"font-style:normal; font-size:90%\">\\(-x\\)<\/span>.<\/p>\n<p>So we have:<\/p>\n<div class=\"examplesentence\">\\(f(-x)=(-x+1)^{2}\\)<\/div>\n<p>\n&nbsp;<br \/>\nWhich we can write as:<\/p>\n<div class=\"examplesentence\">\\(f(-x)=(-x+1)^{2}\\)\\(=(-x+1)(-x+1)\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnd then from here we can FOIL:<\/p>\n<div class=\"examplesentence\">\\(-x\\cdot-x=x^{2}\\)<br \/>\n\\(1\\cdot-x=-x\\)<br \/>\n\\(-x\\cdot1=-x\\)<\/div>\n<p>\n&nbsp;<\/p>\n<div class=\"examplesentence\">\\(1\\cdot1=1\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo we have:<\/p>\n<div class=\"examplesentence\">\\(f(-x)=x^{2}-2x+1\\)<\/div>\n<p>\n&nbsp;<br \/>\nBut remember, to determine even or odd, we have to compare this to our original function. So let&#8217;s expand that.<\/p>\n<p>For this, we have:<\/p>\n<div class=\"examplesentence\">\\(f(x)=(x+1)(x+1)\\)<\/div>\n<p>\n&nbsp;<br \/>\nWhich is:<\/p>\n<div class=\"examplesentence\">\\(x\u22c5x=x^{2}\\)<br \/>\n\\(1\\cdot x=x\\text{ plus }x\\cdot1=x,\\)\\(\\text{ so plus }2x\\)<br \/>\n\\(1\\cdot1=1\\)<br \/>\n\\(f(x)=(x+1)(x+1)\\)\\(=x^{2}+2x+1\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow, when we compare these two functions, we see that only one of the three terms ended up changing signs, so <span style=\"font-style:normal; font-size:90%\">\\(f(x)\\)<\/span> is not odd. And since one term did change sign, <span style=\"font-style:normal; font-size:90%\">\\(f(x)\\neq f(-x)\\)<\/span>, so the function is not even either.<\/p>\n<h2><span id=\"The_Reason_for_Even_and_Odd_Functions\" class=\"m-toc-anchor\"><\/span>The Reason for Even and Odd Functions<\/h2>\n<p>\nNow that we\u2019ve laid a groundwork for understanding even and odd functions, let\u2019s talk about why we call them even and odd in the first place. <\/p>\n<p>Remember that even functions are the same when we evaluate them at <span style=\"font-style:normal; font-size:90%\">\\(+x\\)<\/span> and at <span style=\"font-style:normal; font-size:90%\">\\(-x\\)<\/span>. As we saw earlier, <span style=\"font-style:normal; font-size:90%\">\\(f(x)=x^{2}\\)<\/span> satisfies this property because anytime we square something, a positive value is returned, and therefore the sign of that term doesn\u2019t change even if we plug a negative value in. The same is true when something is raised to the fourth power, or the sixth, and so on.<\/p>\n<p>Notice that constants do not change sign when we evaluate at <span style=\"font-style:normal; font-size:90%\">\\(-x\\)<\/span> either. That\u2019s why we saw that the function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=x^{2}+1\\)<\/span> was still even. As you can see, <strong>an even function will have even exponents<\/strong>.<\/p>\n<p>It may be unsurprising now that <strong>odd functions likewise will have odd exponents<\/strong>! Remember that in order for a function to be odd, all terms must change sign when we evaluate at <span style=\"font-style:normal; font-size:90%\">\\(-x\\)<\/span>. <\/p>\n<p>Clearly, any term with <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span> to the first power will change sign when we plug in a negative value of <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>. In the same way, <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span> to the third power, the fifth power, and so on will all change sign when we plug in a negative value for <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>.<\/p>\n<p>As we mentioned earlier, when a term has an even power of <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>, it will not change sign. That means that an odd function cannot contain any terms with even powers of <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>, and it cannot have any constants.<\/p>\n<p>You may recognize even and odd functions later on in calculus when it comes to dealing with the Taylor expansion.<\/p>\n<h2><span id=\"Review_Questions\" class=\"m-toc-anchor\"><\/span>Review Questions<\/h2>\n<p>\nTime for some practice problems!<\/p>\n<h3><span id=\"Question_1\" class=\"m-toc-anchor\"><\/span>Question #1<\/h3>\n<p>\nBased on this graph, is this function even, odd, or neither?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Even-and-Odd-Functions-4.png\" alt=\"A graph that is not odd or even.\" width=\"777\" height=\"437\" class=\"aligncenter size-full wp-image-89947\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Even-and-Odd-Functions-4.png 1920w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Even-and-Odd-Functions-4-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Even-and-Odd-Functions-4-1024x576.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Even-and-Odd-Functions-4-768x432.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Even-and-Odd-Functions-4-1536x864.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>Neither. This function is not symmetric about the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis, so it is not even. And even though it passes through the origin, it is not odd either because it would not appear the same if we were to rotate the image 180\u00b0.<\/p>\n<h3><span id=\"Question_2\" class=\"m-toc-anchor\"><\/span>Question #2<\/h3>\n<p>\nLet\u2019s try another one.<\/p>\n<p>Is the function <span style=\"font-style:normal; font-size:90%\">\\(f(x)=\\frac{5}{7}x^{3}-2x\\)<\/span> even, odd, or neither?<\/p>\n<p>Let\u2019s look at each term. First, <span style=\"font-style:normal; font-size:90%\">\\(\\frac{5}{7}x^{3}\\)<\/span> has an odd power of <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>, meaning that the sign will change when evaluated at <span style=\"font-style:normal; font-size:90%\">\\(-x\\)<\/span>. Similarly, the second term, <span style=\"font-style:normal; font-size:90%\">\\(-2x\\)<\/span>, has an odd power of <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span> and will also change sign. That means this function is odd!<\/p>\n<h3><span id=\"Question_3\" class=\"m-toc-anchor\"><\/span>Question #3<\/h3>\n<p>\nLet\u2019s finish with a more conceptual question.<\/p>\n<p>We know that some functions may be neither even nor odd, but is it possible for a function to be both even and odd?<\/p>\n<p>Surprisingly, the answer is yes, but only for one function. Can you think of what function that is? Remember that for even functions, <span style=\"font-style:normal; font-size:90%\">\\(f(-x)=f(x)\\)<\/span>, and for odd functions, <span style=\"font-style:normal; font-size:90%\">\\(f(-x)=-f(x)\\)<\/span>. The only way both of these can be satisfied is when <span style=\"font-style:normal; font-size:90%\">\\(f(x)=0\\)<\/span>.<\/p>\n<div class=\"examplesentence\">\\(f(-x)=f(x)\\)<br \/>\n\\(and\\)<br \/>\n\\(f(-x)=-f(x)\\)<\/div>\n<p>\n&nbsp;<br \/>\nAs a quick recap, we can identify even and odd functions in the following ways:<\/p>\n<p>Graphically, even functions are symmetric about the <span style=\"font-style:normal; font-size:90%\">\\(y\\)<\/span>-axis. And they don\u2019t have to pass through the origin. Though, odd functions must pass through the origin, and they will appear the same when viewed from a 180\u00b0 rotation.<\/p>\n<p>Algebraically, even functions are the same when we evaluate at <span style=\"font-style:normal; font-size:90%\">\\(+x\\)<\/span> and at <span style=\"font-style:normal; font-size:90%\">\\(-x\\)<\/span>. Odd functions will change signs across all terms when evaluated at <span style=\"font-style:normal; font-size:90%\">\\(-x\\)<\/span>.<\/p>\n<p>As a shortcut, if a function contains only even exponents of <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span> (and may or may not have constants) then it is even. If a function has no constants and only odd exponents of <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>, then it is odd.<\/p>\n<hr>\n<p>\nNow that we\u2019ve covered everything and run through some examples, you should be pretty comfortable with identifying even and odd functions.<\/p>\n<p>Thanks for watching, and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Even_and_Odd_Function_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Even and Odd Function Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe graph of the function \\(y=f(x)\\) is shown on the coordinate plane below. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Determining-even-and-odd-functions-example-1.svg\" alt=\"A graph of a cubic function with a local maximum near x = -3, a local minimum near x = -1, and increasing steeply for x &gt; 0.\" width=\"1042\" height=\"683\" class=\"aligncenter size-full wp-image-274954\"  role=\"img\" \/><\/p>\n<p>Based on the graph, which of the following statements is true?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">The function is an even function.<\/div><div class=\"PQ\"  id=\"PQ-1-2\">The function is an odd function.<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-3\">The function is neither an even nor an odd function.<\/div><div class=\"PQ\"  id=\"PQ-1-4\">The function is both an even and an odd function.<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The graph of a function is even when it has symmetry about the \\(y\\)-axis. The graph will appear to be mirrored about the \\(y\\)-axis.<\/p>\n<p>In other words, reflecting the part of the graph of the function that lies to the right of the \\(y\\)-axis produces the part of the graph that lies on the left side of the \\(y\\)-axis. This means that any point \\((x,y)\\) on the graph of the function reflects to \\((-x,y)\\) which is also on the graph of the function. Below is the graph of the function and its reflection about the \\(y\\)-axis, shown in red.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Determining-even-and-odd-functions-example-2.svg\" alt=\"A graph of a function with a vertical asymptote at x = 0, showing peaks near x = -2 and x = 2, and the curve colored blue on the left and red on the right of the asymptote.\" width=\"1042\" height=\"683\" class=\"aligncenter size-full wp-image-274957\"  role=\"img\" \/><\/p>\n<p>Notice that the reflected graph of \\(y=f(x)\\) about the \\(y\\)-axis produces an entirely different graph. Thus, the graph of \\(y=f(x)\\) does not have symmetry about the \\(y\\)-axis, so it is not an even function.<\/p>\n<p>The graph of a function is odd when it has symmetry about the origin. The graph will appear to be the same when it is rotated 180\u00b0 about the origin. This means that any point \\((x,y)\\) on the graph of the function reflects to \\((-x,-y)\\) which is also on the graph of the function.<\/p>\n<p>Below is a graph of the function and its reflection about the origin shown in red.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Determining-even-and-odd-functions-example-3.svg\" alt=\"A graph with two cubic polynomial curves: a blue curve on the left dipping and rising, and a red curve on the right rising and falling, both on a grid background.\" width=\"1042\" height=\"683\" class=\"aligncenter size-full wp-image-274960\"  role=\"img\" \/><\/p>\n<p>Notice that the reflected graph of \\(y=f(x)\\) about the origin produces an entirely different graph. Thus, the graph of \\(y=f(x)\\) does not have symmetry about the origin, so is not an odd function. Additionally, another condition to consider when determining if the graph of a function is odd, is that it must pass through the origin.<\/p>\n<p>Therefore, the graph of \\(y=f(x)\\) is neither an even nor an odd function.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich of the following statements best describes the function \\(f(x)=2x-x^3\\)?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">The function is an even function.<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-2\">The function is an odd function.<\/div><div class=\"PQ\"  id=\"PQ-2-3\">The function is neither an even nor an odd function.<\/div><div class=\"PQ\"  id=\"PQ-2-4\">The function is both an even and an odd function.<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>A function, \\(y=f(x)\\), is an even function when replacing the \\(x\\)-value in the function with \\(-x\\) does not change the value of the function. That is, \\(y=f(-x)=f(x)\\).<\/p>\n<p>Replacing \\(x\\) with \\(-x\\) for our function and simplifying, we get:<\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(f(-x)=2(-x)-(-x)^3\\)\\(\\:=-2x-(-x^3)\\)\\(\\:=-2x+x^3\\)\\(\\:=-(2x-x^3)=-f(x)\\)<\/p>\n<p>Since \\(f(-x)=-f(x)\\), the value of the function changes sign when replacing it with \\(-x\\), so the function is not an even function.<\/p>\n<p>A function, \\(y=f(x)\\), is an odd function when replacing the \\(x\\)-value in the function with \\(-x\\) changes the value of the function. That is, \\(y=f(-x)=-f(x)\\). As we saw from above, replacing \\(x\\) with \\(-x\\) for our function produces \\(f(-x)=-f(x)\\), so our function is an odd function.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich of the following statements best describes the function \\(f\\left(x\\right)=x^4-2x^2+1\\)?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-3-1\">The function is an even function.<\/div><div class=\"PQ\"  id=\"PQ-3-2\">The function is an odd function.<\/div><div class=\"PQ\"  id=\"PQ-3-3\">The function is neither an even nor an odd function.<\/div><div class=\"PQ\"  id=\"PQ-3-4\">The function is both an even and an odd function.<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>A function, \\(y=f(x)\\), is an even function when replacing the \\(x\\)-value in the function with \\(-x\\) does not change the value of the function. That is, \\(y=f(-x)=f(x)\\). Replacing \\(x\\) with \\(-x\\) for our function and simplifying, we get:<\/p>\n<p style=\"text-align: center; line-height: 35px\">\\(f(-x)\\)\\(\\:={(-x)}^4-2{(-x)}^2+1\\)\\(\\:=x^4-2x^2+1=f(x)\\)<\/p>\n<p>Since \\(f(-x)=f(x)\\), the value of the function does not change sign when replacing it with \\(-x\\), so the function is an even function.<\/p>\n<p>A function, \\(y=f(x)\\), is an odd function when replacing the \\(x\\)-value in the function with \\(-x\\) changes the value of the function. That is, \\(y=f(-x)=-f(x)\\). As we saw from above, replacing \\(x\\) with \\(-x\\) for our function produces \\(f(-x)=f(x)\\), so our function is not an odd function.<\/p>\n<p>Therefore, the function is only an even function.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nA picture of a cross section of a bowl-shaped skateboard ramp that was taken at a skatepark is shown on the coordinate plane below. Let the cross section of the curved shape of the ramp be the function \\(y=f(x)\\).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Determining-even-and-odd-functions-example-4.svg\" alt=\"A skateboarder is performing a trick on the left edge of a half-pipe shown on a grid with labeled axes.\" width=\"1042\" height=\"683\" class=\"aligncenter size-full wp-image-274963\"  role=\"img\" \/><\/p>\n<p>If we are looking at the bottom of the bowl-shaped ramp, which of the following statements appears to be true about the graph of the function that represents the cross section of the ramp?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-4-1\">The function is an even function.<\/div><div class=\"PQ\"  id=\"PQ-4-2\">The function is an odd function.<\/div><div class=\"PQ\"  id=\"PQ-4-3\">The function is neither an even nor an odd function.<\/div><div class=\"PQ\"  id=\"PQ-4-4\">The function is both an even and an odd function.<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The graph of a function is even when it has symmetry about the \\(y\\)-axis. The graph will appear to be mirrored about the \\(y\\)-axis. Reflecting the part of the graph of the function that lies to the right of the \\(y\\)-axis produces the part of the graph that lies on the left side of the \\(y\\)-axis. This means that any point \\((x,y)\\) on the graph of the function reflects to \\((-x,y)\\) which is also on the graph of the function.<\/p>\n<p>Below is the graph of the ramp and its reflection about the \\(y\\)-axis.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Determining-even-and-odd-functions-example-5.svg\" alt=\"A graph showing a shaded gray area, a blue curved line between points (-1.8, 3.2) and (1.8, 3.2), with red dashed line connecting these points horizontally above the curve.\" width=\"1042\" height=\"683\" class=\"aligncenter size-full wp-image-274966\"  role=\"img\" \/><\/p>\n<p>Notice that reflecting the side of the graph to the right of the \\(y\\)-axis about the \\(y\\)-axis produces the portion of the ramp that is to the left of the \\(y\\)-axis. One such point on the graph of the ramp bears this out and this is true for all points on the graph of the ramp. Thus, the graph the function \\(y=f(x)\\) that represents the skate ramp has symmetry about the \\(y\\)-axis, so it is an even function.<\/p>\n<p>The graph of a function is odd when it has symmetry about the origin. The graph will appear to be the same when it is rotated 180\u00b0 about the origin. This means that any point \\((x,y)\\) on the graph of the function reflects to \\((-x,-y)\\) which is also on the graph of the function. Below is the graph of the ramp and its reflection about the origin.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Determining-even-and-odd-functions-example-6.svg\" alt=\"Graph showing two curved lines, one red and one blue, within a shaded region. Points (-1.8,-3.2) and (1.8,3.2) are marked with black dots and labeled.\" width=\"1042\" height=\"683\" class=\"aligncenter size-full wp-image-274969\"  role=\"img\" \/><\/p>\n<p>Notice that the reflected graph of the ramp about the origin produces an entirely different graph. Thus, the graph of the ramp does not have symmetry about the origin, so is not an odd function. <\/p>\n<p>Therefore, the graph of \\(y=f(x)\\) that represents the skate ramp is only an even function.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nRadio waves are electromagnetic waves that travel at or close to the speed of light. There are many types of radio waves that occur in nature, such as light waves, and ones that are generated artificially with machines. One such artificial wave, called an FM (frequency modulation) radio wave, transmits a carrier signal from a radio station that carries information to your radio\u2019s antenna in which the amplitude of the carrier signal is constant, but the frequency is modulated, or changes.<\/p>\n<p>Below is an example of two cycles, or periods, of an FM carrier wave that has been modulated. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Determining-even-and-odd-functions-example-7.svg\" alt=\"Line graph with multiple peaks and troughs plotted on a Cartesian plane, showing a wave-like pattern intersecting the x-axis several times.\" width=\"1149\" height=\"529\" class=\"aligncenter size-full wp-image-274972\"  role=\"img\" \/><\/p>\n<p>According to the graph of the signal, what type of function is the FM radio wave?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">An even function<\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-2\">An odd function<\/div><div class=\"PQ\"  id=\"PQ-5-3\">Neither an even nor an odd function<\/div><div class=\"PQ\"  id=\"PQ-5-4\">Both an even and an odd function<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The graph of a function is even when it has symmetry about the \\(y\\)-axis. The graph will appear to be mirrored about the \\(y\\)-axis. Reflecting the part of the graph of the function that lies to the right of the \\(y\\)-axis produces the part of the graph that lies on the left side of the \\(y\\)-axis. This means that any point \\((x,y)\\) on the graph of the function reflects to \\((-x,y)\\) which is also on the graph of the function.<\/p>\n<p>Below is a graph of the radio wave and its reflection about the \\(y\\)-axis, shown in red.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Determining-even-and-odd-functions-example-8.svg\" alt=\"Two oscillating curves, one red and one blue, with labeled points: (-3.5, 2.2), (-1.5, -1), (1.5, -1), and (3.5, 2.2) on a grid.\" width=\"1147\" height=\"529\" class=\"aligncenter size-full wp-image-274975\"  role=\"img\" \/><\/p>\n<p>Notice that the reflected graph of the radio wave about the \\(y\\)-axis produces an entirely different graph. Thus, the graph of the radio wave does not have symmetry about the \\(y\\)-axis, so is not an even function.<\/p>\n<p>The graph of a function is odd when it has symmetry about the origin. The graph will appear to be the same when it is rotated 180\u00b0 about the origin. This means that any point \\((x,y)\\) on the graph of the function reflects to \\((-x,-y)\\) which is also on the graph of the function. Below is the graph of the radio wave and its reflected graph about the origin.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/11\/Determining-even-and-odd-functions-example-9.svg\" alt=\"A graph of a fluctuating blue curve with labeled points at (-3.5, 2.2), (-1.5, -1), (1.5, 1), and (3.5, -2.2), connected by dashed red lines.\" width=\"1147\" height=\"528\" class=\"aligncenter size-full wp-image-274951\"  role=\"img\" \/><\/p>\n<p>Notice that reflecting the side of the graph to the right of the \\(y\\)-axis about the origin produces the portion of the ramp that is to the left of the \\(y\\)-axis. Two such points on the graph of the radio wave bear this out, and this is true for all points on the graph of the radio wave. Thus, the graph of the function that represents the radio wave has symmetry about the origin, so it is an odd function.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/algebra-ii\/\">Return to Algebra II Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Algebra II Videos<\/p>\n","protected":false},"author":1,"featured_media":100708,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-86581","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-functions-and-their-graphs-videos","7":"page_category-video-pages-for-study-course-sidebar-ad","8":"page_type-video","9":"content_type-practice-questions","10":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/86581","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=86581"}],"version-history":[{"count":8,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/86581\/revisions"}],"predecessor-version":[{"id":280466,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/86581\/revisions\/280466"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/100708"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=86581"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}