{"id":148,"date":"2023-09-20T22:47:53","date_gmt":"2023-09-20T22:47:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/symmetry-of-functions\/"},"modified":"2024-08-30T19:47:07","modified_gmt":"2024-08-30T19:47:07","slug":"review-of-functions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/review-of-functions-learn-it-4\/","title":{"raw":"Review of Functions: Learn it 4","rendered":"Review of Functions: Learn it 4"},"content":{"raw":"<h2 class=\"entry-title\">Symmetry of Functions<\/h2>\r\n<p>Function graphs often exhibit symmetry, a feature that can simplify understanding their behavior.<\/p>\r\n<p>Symmetry about the [latex]y[\/latex]-axis means that mirroring the graph over the [latex]y[\/latex]-axis results in the same graph, indicating an even function where [latex]f(x)=f(\u2212x)[\/latex]. For instance, [latex]f(x)=x^4\u22122x^2\u22123[\/latex] is even because both sides of the [latex]y[\/latex]-axis mirror each other.<\/p>\r\n<p>Symmetry about the origin implies that rotating the graph [latex]180[\/latex] degrees around the origin leaves the graph unchanged. This is characteristic of odd functions, satisfying [latex]f(\u2212x)=\u2212f(x)[\/latex]. Take [latex]f(x)=x^3\u22124x[\/latex] as an example; it's odd because rotating its graph doesn\u2019t alter it.<\/p>\r\n<section class=\"textbox proTip\">\r\n<p>Algebraically, you can check for y-axis symmetry by seeing if [latex]f(\u2212x)[\/latex] equals [latex]f(x)[\/latex], and for origin symmetry by checking if [latex]f(\u2212x)[\/latex] equals [latex]\u2212f(x)[\/latex].<\/p>\r\n<\/section>\r\n<p><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\" \/><\/p>\r\n<p>Figure 13. (a) A graph that is symmetric about the [latex]y[\/latex]-axis. (b) A graph that is symmetric about the origin.<\/p>\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? It becomes straightforward to identify symmetry in functions once we determine if they are even or odd. Even functions are symmetric about the y-axis, whereas odd functions exhibit symmetry about the origin.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>even and odd functions<\/h3>\r\n<ul>\r\n\t<li>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 <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><\/li>\r\n\t<li>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.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572169681\">Determine whether each of the following functions is even, odd, or neither.<\/p>\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<p>[reveal-answer q=\"fs-id1170572477853\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572477853\"]<\/p>\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<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\r\nWatch the following video to see the worked solution this example.<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>\u00a0 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. You can view the transcript for this segmented clip <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.1ReviewofFunctions1906to2032_transcript.txt\" target=\"_blank\" rel=\"noopener\"> using this link<\/a> (opens in new window).\r\n\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]197087[\/ohm_question]<\/p>\r\n<\/section>\r\n<h3>Absolute Value Function<\/h3>\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<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<p>&nbsp;<\/p>\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 can conclude that the range of the absolute value function is [latex]\\{y|y\\ge 0\\}[\/latex].<\/p>\r\n<p>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\r\n<section class=\"textbox example\">\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<p style=\"text-align: left;\">[reveal-answer q=\"fs-id1170572548722\"]Show Solution[\/reveal-answer]<br \/>\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\n<p>The range of this function is [latex]\\{y|y\\ge 4\\}[\/latex].<\/p>\r\n\r\nWatch the following video to see the worked solution to this example.<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>\u00a0 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\n<p style=\"text-align: left;\">You can view the transcript for this video <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.1ReviewofFunctions2035to2093_transcript.txt\" target=\"_blank\" rel=\"noopener\"> using this link<\/a> (opens in new window).<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<h2 class=\"entry-title\">Symmetry of Functions<\/h2>\n<p>Function graphs often exhibit symmetry, a feature that can simplify understanding their behavior.<\/p>\n<p>Symmetry about the [latex]y[\/latex]-axis means that mirroring the graph over the [latex]y[\/latex]-axis results in the same graph, indicating an even function where [latex]f(x)=f(\u2212x)[\/latex]. For instance, [latex]f(x)=x^4\u22122x^2\u22123[\/latex] is even because both sides of the [latex]y[\/latex]-axis mirror each other.<\/p>\n<p>Symmetry about the origin implies that rotating the graph [latex]180[\/latex] degrees around the origin leaves the graph unchanged. This is characteristic of odd functions, satisfying [latex]f(\u2212x)=\u2212f(x)[\/latex]. Take [latex]f(x)=x^3\u22124x[\/latex] as an example; it&#8217;s odd because rotating its graph doesn\u2019t alter it.<\/p>\n<section class=\"textbox proTip\">\n<p>Algebraically, you can check for y-axis symmetry by seeing if [latex]f(\u2212x)[\/latex] equals [latex]f(x)[\/latex], and for origin symmetry by checking if [latex]f(\u2212x)[\/latex] equals [latex]\u2212f(x)[\/latex].<\/p>\n<\/section>\n<p><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>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<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? It becomes straightforward to identify symmetry in functions once we determine if they are even or odd. Even functions are symmetric about the y-axis, whereas odd functions exhibit symmetry about the origin.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>even and odd functions<\/h3>\n<ul>\n<li>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 <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><\/li>\n<li>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.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">\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<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572477853\">Show Solution<\/button><\/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<p>Watch the following video to see the worked solution this example.<\/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>\u00a0 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. You can view the transcript for this segmented clip <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.1ReviewofFunctions1906to2032_transcript.txt\" target=\"_blank\" rel=\"noopener\"> using this link<\/a> (opens in new window).<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm197087\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=197087&theme=lumen&iframe_resize_id=ohm197087&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<h3>Absolute Value Function<\/h3>\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 can conclude that the range of the absolute value function is [latex]\\{y|y\\ge 0\\}[\/latex].<\/p>\n<p>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<figure style=\"width: 325px\" 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\" \/><figcaption class=\"wp-caption-text\">Figure 14. The graph of [latex]f(x)=|x|[\/latex] is symmetric about the [latex]y[\/latex]-axis.<\/figcaption><\/figure>\n<section class=\"textbox example\">\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 style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572548722\">Show Solution<\/button><\/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<p>Watch the following video to see the worked solution to this example.<\/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>\u00a0 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 transcript for this video <a href=\"https:\/\/oerfiles.s3-us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/1.1ReviewofFunctions2035to2093_transcript.txt\" target=\"_blank\" rel=\"noopener\"> using this link<\/a> (opens in new window).<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":6,"menu_order":7,"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\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":143,"module-header":"- Select Header -","content_attributions":[{"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":""}],"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\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/148"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":21,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/148\/revisions"}],"predecessor-version":[{"id":4638,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/148\/revisions\/4638"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/143"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/148\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=148"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=148"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=148"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=148"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}