{"id":96,"date":"2021-02-03T20:54:01","date_gmt":"2021-02-03T20:54:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=96"},"modified":"2022-03-11T21:35:39","modified_gmt":"2022-03-11T21:35:39","slug":"symmetry-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/symmetry-of-functions\/","title":{"raw":"Symmetry of Functions","rendered":"Symmetry of Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Describe the symmetry properties of a function<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1170572213225\">The graphs of certain functions have symmetry properties that help us understand the function and the shape of its graph. For example, consider the function [latex]f(x)=x^4-2x^2-3[\/latex] shown in Figure 13(a). If we take the part of the curve that lies to the right of the [latex]y[\/latex]-axis and flip it over the [latex]y[\/latex]-axis, it lays exactly on top of the curve to the left of the [latex]y[\/latex]-axis. In this case, we say the function has<strong> symmetry about the [latex]y[\/latex]-axis<\/strong>. On the other hand, consider the function [latex]f(x)=x^3-4x[\/latex] shown in Figure 13(b). If we take the graph and rotate it 180\u00b0 about the origin, the new graph will look exactly the same. In this case, we say the function has <strong>symmetry about the origin.<\/strong><\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202138\/CNX_Calc_Figure_01_01_012.jpg\" alt=\"An image of two graphs. The first graph is labeled \u201c(a), symmetry about the y-axis\u201d and is of the curved function \u201cf(x) = (x to the 4th) - 2(x squared) - 3\u201d. The x axis runs from -3 to 4 and the y axis runs from -4 to 5. This function decreases until it hits the point (-1, -4), which is minimum of the function. Then the graph increases to the point (0,3), which is a local maximum. Then the the graph decreases until it hits the point (1, -4), before it increases again. The second graph is labeled \u201c(b), symmetry about the origin\u201d and is of the curved function \u201cf(x) = x cubed - 4x\u201d. The x axis runs from -3 to 4 and the y axis runs from -4 to 5. The graph of the function starts at the x intercept at (-2, 0) and increases until the approximate point of (-1.2, 3.1). The function then decreases, passing through the origin, until it hits the approximate point of (1.2, -3.1). The function then begins to increase again and has another x intercept at (2, 0).\" width=\"731\" height=\"426\" \/> Figure 13. (a) A graph that is symmetric about the [latex]y[\/latex]-axis. (b) A graph that is symmetric about the origin.[\/caption]\r\n<p id=\"fs-id1170572173116\">If we are given the graph of a function, it is easy to see whether the graph has one of these symmetry properties. But without a graph, how can we determine algebraically whether a function [latex]f[\/latex] has symmetry? Looking at Figure 13(a) again, we see that since [latex]f[\/latex] is symmetric about the [latex]y[\/latex]-axis, if the point [latex](x,y)[\/latex] is on the graph, the point [latex](\u2212x,y)[\/latex] is on the graph. In other words, [latex]f(\u2212x)=f(x)[\/latex]. If a function [latex]f[\/latex] has this property, we say [latex]f[\/latex] is an <strong>even function<\/strong>, which has symmetry about the [latex]y[\/latex]-axis. For example, [latex]f(x)=x^2[\/latex] is even because<\/p>\r\n\r\n<div id=\"fs-id1170572552270\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(\u2212x)=(\u2212x)^2=x^2=f(x)[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572552328\">In contrast, looking at Figure 13(b) again, if a function [latex]f[\/latex] is symmetric about the origin, then whenever the point [latex](x,y)[\/latex] is on the graph, the point [latex](\u2212x,\u2212y)[\/latex] is also on the graph. In other words, [latex]f(\u2212x)=\u2212f(x)[\/latex]. If [latex]f[\/latex] has this property, we say [latex]f[\/latex] is an <strong>odd function<\/strong>, which has symmetry about the origin. For example, [latex]f(x)=x^3[\/latex] is odd because<\/p>\r\n\r\n<div id=\"fs-id1170572548910\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(\u2212x)=(\u2212x)^3=\u2212x^3=\u2212f(x)[\/latex].<\/div>\r\n&nbsp;\r\n<div>\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\"><span style=\"font-size: 1rem; text-align: initial;\">Definition<\/span><\/h3>\r\n\r\n<hr \/>\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">If [latex]f(-x)=f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex], then [latex]f[\/latex] is an even function. An <\/span><strong style=\"font-size: 1rem; text-align: initial;\">even function<\/strong><span style=\"font-size: 1rem; text-align: initial;\"> is symmetric about the [latex]y[\/latex]-axis.<\/span>\r\n<p id=\"fs-id1170572169613\">If [latex]f(\u2212x)=\u2212f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex], then [latex]f[\/latex] is an odd function. An <strong>odd function<\/strong> is symmetric about the origin.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170572169671\" class=\"textbox exercises\">\r\n<h3>Example: Even and Odd Functions<\/h3>\r\n<p id=\"fs-id1170572169681\">Determine whether each of the following functions is even, odd, or neither.<\/p>\r\n\r\n<ol id=\"fs-id1170572169684\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]f(x)=-5x^4+7x^2-2[\/latex]<\/li>\r\n \t<li>[latex]f(x)=2x^5-4x+5[\/latex]<\/li>\r\n \t<li>[latex]f(x)=\\dfrac{3x}{x^2+1}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1170572477853\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572477853\"]\r\n<p id=\"fs-id1170572477853\">To determine whether a function is even or odd, we evaluate [latex]f(\u2212x)[\/latex] and compare it to [latex]f(x)[\/latex] and [latex]\u2212f(x)[\/latex].<\/p>\r\n\r\n<ol id=\"fs-id1170572477904\" style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]f(\u2212x)=-5(\u2212x)^4+7(\u2212x)^2-2=-5x^4+7x^2-2=f(x)[\/latex]. Therefore, [latex]f[\/latex] is even.<\/li>\r\n \t<li>[latex]f(\u2212x)=2(\u2212x)^5-4(\u2212x)+5=-2x^5+4x+5[\/latex]. Now, [latex]f(\u2212x)\\ne f(x)[\/latex]. Furthermore, noting that [latex]\u2212f(x)=-2x^5+4x-5[\/latex], we see that [latex]f(\u2212x)\\ne \u2212f(x)[\/latex]. Therefore, [latex]f[\/latex] is neither even nor odd.<\/li>\r\n \t<li>[latex]f(\u2212x)=\\frac{3(\u2212x)}{((\u2212x)^2+1)}=\\frac{-3x}{(x^2+1)}=\u2212\\left[\\frac{3x}{(x^2+1)}\\right]=\u2212f(x)[\/latex]. Therefore, [latex]f[\/latex] is odd.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to Example: Even and Odd Functions[\/caption]\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/qL2tyJhmrkg?controls=0&amp;start=1906&amp;end=2032&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>&nbsp;\r\n\r\n[reveal-answer q=\"266833\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266833\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.1ReviewofFunctions1906to2032_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"1.1 Review of Functions\" here (opens in new window)<\/a>.[\/hidden-answer]\r\n<div id=\"fs-id1170572478816\" class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170572478824\">Determine whether [latex]f(x)=4x^3-5x[\/latex] is even, odd, or neither.<\/p>\r\n&nbsp;\r\n\r\n[reveal-answer q=\"935578\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"935578\"]\r\n\r\nCompare [latex]f(\u2212x)[\/latex] with [latex]f(x)[\/latex] and [latex]\u2212f(x)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"fs-id1170572547379\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572547379\"]\r\n<p id=\"fs-id1170572547379\">[latex]f(x)[\/latex] is odd.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p id=\"fs-id1170572547400\">One symmetric function that arises frequently is the <strong>absolute value function<\/strong>, written as [latex]|x|[\/latex]. The absolute value function is defined as<\/p>\r\n\r\n<div id=\"fs-id1170572547420\" class=\"equation\" style=\"text-align: center;\">[latex]f(x)=\\begin{cases} x, &amp; x \\ge 0 \\\\ -x, &amp; x &lt; 0 \\end{cases}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1170572547473\">Some students describe this function by stating that it \u201cmakes everything positive.\u201d By the definition of the absolute value function, we see that if [latex]x&lt;0[\/latex], then [latex]|x|=\u2212x&gt;0[\/latex], and if [latex]x&gt;0[\/latex], then [latex]|x|=x&gt;0[\/latex]. However, for [latex]x=0, \\, |x|=0[\/latex]. Therefore, it is more accurate to say that for all nonzero inputs, the output is positive, but if [latex]x=0[\/latex], the output [latex]|x|=0[\/latex]. We conclude that the range of the absolute value function is [latex]\\{y|y\\ge 0\\}[\/latex]. In Figure 14, we see that the absolute value function is symmetric about the [latex]y[\/latex]-axis and is therefore an even function.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"325\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202141\/CNX_Calc_Figure_01_01_013.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -4 to 4. The graph is of the function \u201cf(x) = absolute value of x\u201d. The graph starts at the point (-3, 3) and decreases in a straight line until it hits the origin. Then the graph increases in a straight line until it hits the point (3, 3).\" width=\"325\" height=\"350\" \/> Figure 14. The graph of [latex]f(x)=|x|[\/latex] is symmetric about the [latex]y[\/latex]-axis.[\/caption]\r\n<div class=\"wp-caption-text\"><\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Working with the Absolute Value Function<\/h3>\r\n<p style=\"text-align: left;\">Find the domain and range of the function [latex]f(x)=2|x-3|+4[\/latex].<\/p>\r\n&nbsp;\r\n<p style=\"text-align: left;\">[reveal-answer q=\"fs-id1170572548722\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572548722\"]<\/p>\r\n<p style=\"text-align: left;\">Since the absolute value function is defined for all real numbers, the domain of this function is [latex](\u2212\\infty ,\\infty )[\/latex]. Since [latex]|x-3|\\ge 0[\/latex] for all [latex]x[\/latex], the function [latex]f(x)=2|x-3|+4\\ge 4[\/latex]. Therefore, the range is, at most, the set [latex]\\{y|y\\ge 4\\}[\/latex]. To see that the range is, in fact, this whole set, we need to show that for [latex]y\\ge 4[\/latex] there exists a real number [latex]x[\/latex] such that<\/p>\r\n<p style=\"text-align: center;\">[latex]2|x-3|+4=y[\/latex].<\/p>\r\n<p style=\"text-align: left;\">A real number [latex]x[\/latex] satisfies this equation as long as<\/p>\r\n<p style=\"text-align: center;\">[latex]|x-3|=\\frac{1}{2}(y-4)[\/latex].<\/p>\r\n<p style=\"text-align: left;\">Since [latex]y\\ge 4[\/latex], we know [latex]y-4\\ge 0[\/latex], and thus the right-hand side of the equation is nonnegative, so it is possible that there is a solution. Furthermore,<\/p>\r\n<p style=\"text-align: center;\">[latex]|x-3|= \\begin{cases} x-3, &amp; \\text{ if } \\, x \\ge 3 \\\\ -(x-3), &amp; \\text{ if } \\, x &lt; 3 \\end{cases}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Therefore, we see there are two solutions:<\/p>\r\n<p style=\"text-align: center;\">[latex]x=\\pm\\frac{1}{2}(y-4)+3[\/latex].<\/p>\r\nThe range of this function is [latex]\\{y|y\\ge 4\\}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to Example: Working with the Absolute Value Function[\/caption]\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/qL2tyJhmrkg?controls=0&amp;start=2035&amp;end=2093&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>&nbsp;\r\n\r\n&nbsp;\r\n<p style=\"text-align: left;\">[reveal-answer q=\"266834\"]Closed Captioning and Transcript Information for Video[\/reveal-answer]\r\n[hidden-answer a=\"266834\"]For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\n<p style=\"text-align: left;\">You can view the <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.1ReviewofFunctions2035to2093_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"1.1 Review of Functions\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\r\n&nbsp;\r\n<div id=\"fs-id1170572548614\" class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1170572548622\">For the function [latex]f(x)=|x+2|-4[\/latex], find the domain and range.<\/p>\r\n&nbsp;\r\n\r\n[reveal-answer q=\"882365\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"882365\"]\r\n<p id=\"fs-id1165043422402\">[latex]|x+2|\\ge 0[\/latex] for all real numbers [latex]x[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n[reveal-answer q=\"fs-id1170572176864\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572176864\"]\r\n<p id=\"fs-id1170572176864\">Domain = [latex](\u2212\\infty ,\\infty )[\/latex], range = [latex]\\{y|y\\ge -4\\}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]197087[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Describe the symmetry properties of a function<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1170572213225\">The graphs of certain functions have symmetry properties that help us understand the function and the shape of its graph. For example, consider the function [latex]f(x)=x^4-2x^2-3[\/latex] shown in Figure 13(a). If we take the part of the curve that lies to the right of the [latex]y[\/latex]-axis and flip it over the [latex]y[\/latex]-axis, it lays exactly on top of the curve to the left of the [latex]y[\/latex]-axis. In this case, we say the function has<strong> symmetry about the [latex]y[\/latex]-axis<\/strong>. On the other hand, consider the function [latex]f(x)=x^3-4x[\/latex] shown in Figure 13(b). If we take the graph and rotate it 180\u00b0 about the origin, the new graph will look exactly the same. In this case, we say the function has <strong>symmetry about the origin.<\/strong><\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202138\/CNX_Calc_Figure_01_01_012.jpg\" alt=\"An image of two graphs. The first graph is labeled \u201c(a), symmetry about the y-axis\u201d and is of the curved function \u201cf(x) = (x to the 4th) - 2(x squared) - 3\u201d. The x axis runs from -3 to 4 and the y axis runs from -4 to 5. This function decreases until it hits the point (-1, -4), which is minimum of the function. Then the graph increases to the point (0,3), which is a local maximum. Then the the graph decreases until it hits the point (1, -4), before it increases again. The second graph is labeled \u201c(b), symmetry about the origin\u201d and is of the curved function \u201cf(x) = x cubed - 4x\u201d. The x axis runs from -3 to 4 and the y axis runs from -4 to 5. The graph of the function starts at the x intercept at (-2, 0) and increases until the approximate point of (-1.2, 3.1). The function then decreases, passing through the origin, until it hits the approximate point of (1.2, -3.1). The function then begins to increase again and has another x intercept at (2, 0).\" width=\"731\" height=\"426\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 13. (a) A graph that is symmetric about the [latex]y[\/latex]-axis. (b) A graph that is symmetric about the origin.<\/p>\n<\/div>\n<p id=\"fs-id1170572173116\">If we are given the graph of a function, it is easy to see whether the graph has one of these symmetry properties. But without a graph, how can we determine algebraically whether a function [latex]f[\/latex] has symmetry? Looking at Figure 13(a) again, we see that since [latex]f[\/latex] is symmetric about the [latex]y[\/latex]-axis, if the point [latex](x,y)[\/latex] is on the graph, the point [latex](\u2212x,y)[\/latex] is on the graph. In other words, [latex]f(\u2212x)=f(x)[\/latex]. If a function [latex]f[\/latex] has this property, we say [latex]f[\/latex] is an <strong>even function<\/strong>, which has symmetry about the [latex]y[\/latex]-axis. For example, [latex]f(x)=x^2[\/latex] is even because<\/p>\n<div id=\"fs-id1170572552270\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(\u2212x)=(\u2212x)^2=x^2=f(x)[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572552328\">In contrast, looking at Figure 13(b) again, if a function [latex]f[\/latex] is symmetric about the origin, then whenever the point [latex](x,y)[\/latex] is on the graph, the point [latex](\u2212x,\u2212y)[\/latex] is also on the graph. In other words, [latex]f(\u2212x)=\u2212f(x)[\/latex]. If [latex]f[\/latex] has this property, we say [latex]f[\/latex] is an <strong>odd function<\/strong>, which has symmetry about the origin. For example, [latex]f(x)=x^3[\/latex] is odd because<\/p>\n<div id=\"fs-id1170572548910\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f(\u2212x)=(\u2212x)^3=\u2212x^3=\u2212f(x)[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<div>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\"><span style=\"font-size: 1rem; text-align: initial;\">Definition<\/span><\/h3>\n<hr \/>\n<p><span style=\"font-size: 1rem; text-align: initial;\">If [latex]f(-x)=f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex], then [latex]f[\/latex] is an even function. An <\/span><strong style=\"font-size: 1rem; text-align: initial;\">even function<\/strong><span style=\"font-size: 1rem; text-align: initial;\"> is symmetric about the [latex]y[\/latex]-axis.<\/span><\/p>\n<p id=\"fs-id1170572169613\">If [latex]f(\u2212x)=\u2212f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex], then [latex]f[\/latex] is an odd function. An <strong>odd function<\/strong> is symmetric about the origin.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1170572169671\" class=\"textbox exercises\">\n<h3>Example: Even and Odd Functions<\/h3>\n<p id=\"fs-id1170572169681\">Determine whether each of the following functions is even, odd, or neither.<\/p>\n<ol id=\"fs-id1170572169684\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]f(x)=-5x^4+7x^2-2[\/latex]<\/li>\n<li>[latex]f(x)=2x^5-4x+5[\/latex]<\/li>\n<li>[latex]f(x)=\\dfrac{3x}{x^2+1}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572477853\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572477853\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572477853\">To determine whether a function is even or odd, we evaluate [latex]f(\u2212x)[\/latex] and compare it to [latex]f(x)[\/latex] and [latex]\u2212f(x)[\/latex].<\/p>\n<ol id=\"fs-id1170572477904\" style=\"list-style-type: lower-alpha;\">\n<li>[latex]f(\u2212x)=-5(\u2212x)^4+7(\u2212x)^2-2=-5x^4+7x^2-2=f(x)[\/latex]. Therefore, [latex]f[\/latex] is even.<\/li>\n<li>[latex]f(\u2212x)=2(\u2212x)^5-4(\u2212x)+5=-2x^5+4x+5[\/latex]. Now, [latex]f(\u2212x)\\ne f(x)[\/latex]. Furthermore, noting that [latex]\u2212f(x)=-2x^5+4x-5[\/latex], we see that [latex]f(\u2212x)\\ne \u2212f(x)[\/latex]. Therefore, [latex]f[\/latex] is neither even nor odd.<\/li>\n<li>[latex]f(\u2212x)=\\frac{3(\u2212x)}{((\u2212x)^2+1)}=\\frac{-3x}{(x^2+1)}=\u2212\\left[\\frac{3x}{(x^2+1)}\\right]=\u2212f(x)[\/latex]. Therefore, [latex]f[\/latex] is odd.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Even and Odd Functions<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/qL2tyJhmrkg?controls=0&amp;start=1906&amp;end=2032&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266833\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266833\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.1ReviewofFunctions1906to2032_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;1.1 Review of Functions&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<div id=\"fs-id1170572478816\" class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170572478824\">Determine whether [latex]f(x)=4x^3-5x[\/latex] is even, odd, or neither.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q935578\">Hint<\/span><\/p>\n<div id=\"q935578\" class=\"hidden-answer\" style=\"display: none\">\n<p>Compare [latex]f(\u2212x)[\/latex] with [latex]f(x)[\/latex] and [latex]\u2212f(x)[\/latex].<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572547379\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572547379\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572547379\">[latex]f(x)[\/latex] is odd.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170572547400\">One symmetric function that arises frequently is the <strong>absolute value function<\/strong>, written as [latex]|x|[\/latex]. The absolute value function is defined as<\/p>\n<div id=\"fs-id1170572547420\" class=\"equation\" style=\"text-align: center;\">[latex]f(x)=\\begin{cases} x, & x \\ge 0 \\\\ -x, & x < 0 \\end{cases}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1170572547473\">Some students describe this function by stating that it \u201cmakes everything positive.\u201d By the definition of the absolute value function, we see that if [latex]x<0[\/latex], then [latex]|x|=\u2212x>0[\/latex], and if [latex]x>0[\/latex], then [latex]|x|=x>0[\/latex]. However, for [latex]x=0, \\, |x|=0[\/latex]. Therefore, it is more accurate to say that for all nonzero inputs, the output is positive, but if [latex]x=0[\/latex], the output [latex]|x|=0[\/latex]. We conclude that the range of the absolute value function is [latex]\\{y|y\\ge 0\\}[\/latex]. In Figure 14, we see that the absolute value function is symmetric about the [latex]y[\/latex]-axis and is therefore an even function.<\/p>\n<div style=\"width: 335px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202141\/CNX_Calc_Figure_01_01_013.jpg\" alt=\"An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -4 to 4. The graph is of the function \u201cf(x) = absolute value of x\u201d. The graph starts at the point (-3, 3) and decreases in a straight line until it hits the origin. Then the graph increases in a straight line until it hits the point (3, 3).\" width=\"325\" height=\"350\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 14. The graph of [latex]f(x)=|x|[\/latex] is symmetric about the [latex]y[\/latex]-axis.<\/p>\n<\/div>\n<div class=\"wp-caption-text\"><\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Working with the Absolute Value Function<\/h3>\n<p style=\"text-align: left;\">Find the domain and range of the function [latex]f(x)=2|x-3|+4[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572548722\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572548722\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">Since the absolute value function is defined for all real numbers, the domain of this function is [latex](\u2212\\infty ,\\infty )[\/latex]. Since [latex]|x-3|\\ge 0[\/latex] for all [latex]x[\/latex], the function [latex]f(x)=2|x-3|+4\\ge 4[\/latex]. Therefore, the range is, at most, the set [latex]\\{y|y\\ge 4\\}[\/latex]. To see that the range is, in fact, this whole set, we need to show that for [latex]y\\ge 4[\/latex] there exists a real number [latex]x[\/latex] such that<\/p>\n<p style=\"text-align: center;\">[latex]2|x-3|+4=y[\/latex].<\/p>\n<p style=\"text-align: left;\">A real number [latex]x[\/latex] satisfies this equation as long as<\/p>\n<p style=\"text-align: center;\">[latex]|x-3|=\\frac{1}{2}(y-4)[\/latex].<\/p>\n<p style=\"text-align: left;\">Since [latex]y\\ge 4[\/latex], we know [latex]y-4\\ge 0[\/latex], and thus the right-hand side of the equation is nonnegative, so it is possible that there is a solution. Furthermore,<\/p>\n<p style=\"text-align: center;\">[latex]|x-3|= \\begin{cases} x-3, & \\text{ if } \\, x \\ge 3 \\\\ -(x-3), & \\text{ if } \\, x < 3 \\end{cases}[\/latex]<\/p>\n<p style=\"text-align: left;\">Therefore, we see there are two solutions:<\/p>\n<p style=\"text-align: center;\">[latex]x=\\pm\\frac{1}{2}(y-4)+3[\/latex].<\/p>\n<p>The range of this function is [latex]\\{y|y\\ge 4\\}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Working with the Absolute Value Function<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/qL2tyJhmrkg?controls=0&amp;start=2035&amp;end=2093&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q266834\">Closed Captioning and Transcript Information for Video<\/span><\/p>\n<div id=\"q266834\" class=\"hidden-answer\" style=\"display: none\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p style=\"text-align: left;\">You can view the <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.1ReviewofFunctions2035to2093_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;1.1 Review of Functions&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div id=\"fs-id1170572548614\" class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1170572548622\">For the function [latex]f(x)=|x+2|-4[\/latex], find the domain and range.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q882365\">Hint<\/span><\/p>\n<div id=\"q882365\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043422402\">[latex]|x+2|\\ge 0[\/latex] for all real numbers [latex]x[\/latex].<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1170572176864\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1170572176864\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572176864\">Domain = [latex](\u2212\\infty ,\\infty )[\/latex], range = [latex]\\{y|y\\ge -4\\}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm197087\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=197087&theme=oea&iframe_resize_id=ohm197087&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-96\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>1.1 Review of Functions. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"1.1 Review of Functions\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-96","chapter","type-chapter","status-publish","hentry"],"part":21,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/96","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":37,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/96\/revisions"}],"predecessor-version":[{"id":4974,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/96\/revisions\/4974"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/21"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/96\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=96"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=96"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=96"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=96"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}