{"id":154,"date":"2019-08-15T17:11:52","date_gmt":"2019-08-15T17:11:52","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=154"},"modified":"2019-11-30T21:34:37","modified_gmt":"2019-11-30T21:34:37","slug":"functions-and-transformation-of-functions","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/precalculus\/functions-and-transformation-of-functions\/","title":{"rendered":"Functions and Transformation of Functions"},"content":{"rendered":"\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n\n\n<title>Functions and Transformations of Functions &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>We will review some of the important concepts dealing with functions and transformations of functions.  Most likely you have encountered each of these ideas previously, but here we will tie the concepts together. <\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Definition of a Function<\/h4>\n\n\n\n<p>\n\nLet $A$ and $B$ be sets.\n\n<\/p>\n\n\n\n<p>\nA <b>function<\/b> $F:A\\to B$ is a relation that assigns to each\n$x\\in A$ a unique $y\\in B$.  We write $y=f(x)$ and call $y$ the\n<b>value of $f$ at $x$<\/b> or the <b>image of $x$ under $f$<\/b>.\nWe also say that $f$ <b>maps<\/b> $x$ to $y$.\n\n<\/p>\n\n\n\n<p>\nThe set $A$ is called the <b>domain<\/b> of $f$.  The set of all\npossible values of $f(x)$ in $B$ is called the <b>range<\/b> of $f$.\nHere, we will only consider real-valued functions of a real variable,\nso $A$ and $B$ will both be subsets of the real numbers.  If $A$ is\nleft unspecified, we will assume it to be the largest set of real\nnumbers such that for all $x\\in A$, $f(x)$ is real.\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Examples<\/h6>\n\n\n\n<p> Open the menu to see the graph of that function as well as its domain and range. <\/p>\n\n\n<style>.kt-accordion-id_b5fa99-99 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_b5fa99-99 .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_b5fa99-99 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_b5fa99-99 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_b5fa99-99 .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_b5fa99-99 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_b5fa99-99 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_b5fa99-99:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_b5fa99-99 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_b5fa99-99 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_b5fa99-99 kt-accordion-has-5-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_f9bc74-e0\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">$f(x)=x^2$, $x$ real.  <\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"332\" height=\"198\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/trans_figure1.gif?resize=332%2C198&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-404\"\/><\/figure><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-2 kt-pane_9b84b9-a6\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\"> $f(x)=\\sin x$, $x$ real. <\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"357\" height=\"196\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/trans_figure2.gif?resize=357%2C196&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-405\"\/><\/figure><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-3 kt-pane_33ac58-b5\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\"> $f(x)=\\left\\{ \\begin{array}{rl} -1, &amp; x &lt; 0\\\\ 2, &amp; x\\geq 0 \\end{array}\\right.$   <\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"327\" height=\"203\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/trans_figure3.gif?resize=327%2C203&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-406\"\/><\/figure><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-4 kt-pane_ae1451-ff\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\"> $\\displaystyle f(x)=\\frac{1}{x+1}$.   <\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"377\" height=\"206\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/trans_figure4.gif?resize=377%2C206&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-407\"\/><\/figure><\/div>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-5 kt-pane_2acfe1-88\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\"> $f(x)=\\sqrt{x-3}$.  <\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<div class=\"wp-block-image\"><figure class=\"aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"326\" height=\"201\" src=\"https:\/\/i0.wp.com\/math.hmc.edu\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/trans_figure5.gif?resize=326%2C201&#038;ssl=1\" alt=\"Follow the image link for a complete description of the image\" class=\"wp-image-408\"\/><\/figure><\/div>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Even\/Odd Functions<\/h4>\n\n\n\n<p>\n\nA function $f:A\\to B$ is said to be <b>even<\/b> if and only if\n\\[f(-x)=f(x)\\quad {\\small\\textrm{for all }} x\\in A\\]\nand is said to be <b>odd<\/b> if and only if\n\\[f(-x)=-f(x)\\quad {\\small\\textrm{for all }} x\\in A.\\]\nMost functions are neither even nor odd.  \n\n<\/p>\n\n\n\n<p> The graph of an even function is symmetric about the <em>y<\/em>-axis, while the graph of an odd function is  symmetric about the origin. <\/p>\n\n\n<style>.kt-accordion-id_5282e5-50 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_5282e5-50 .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_5282e5-50 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_5282e5-50:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_5282e5-50:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_5282e5-50:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_5282e5-50:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_5282e5-50:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_5282e5-50 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_5282e5-50 .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_5282e5-50:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_5282e5-50:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_5282e5-50:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_5282e5-50:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_5282e5-50:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_5282e5-50:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_5282e5-50:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_5282e5-50:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_5282e5-50:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_5282e5-50:not( .kt-accodion-icon-style-basic ):not( 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.kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_5282e5-50 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_5282e5-50 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_5282e5-50 kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_283421-17\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Symmetry about the y-axis<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n<title>Symmetry about the y-axis<\/title>\n<center><font size=\"+2\">\nSymmetry about the $y$-axis\n<\/font><\/center>\n\n\n\n<p>\n<!------------------------>\n\nThe graph of $y=f(x)$ is <b>symmetric about the $y$-axis<\/b> if and\nonly if\n\\[f(-x)=f(x)\\quad {\\small\\textrm{for all }}x{\\small\\textrm{ in the domain.}}\\]\nHere are three intuitive ways to view this symmetry:\n<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li> Positive and negative values of $x$ yield the same results. <br><br> <\/li><li> The left and right &#8220;sides&#8221; of the graph are &#8220;mirror images&#8221; of each other. <br><br> <\/li><li> If you reflect the curve about the $y$-axis, the graph is unchanged.<\/li><\/ul>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-2 kt-pane_a62ca1-ac\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Symmetry about the Origin<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n<title>Symmetry about the Origin<\/title>\n<center><font size=\"+2\">\nSymmetry about the Origin\n<\/font><\/center>\n\n\n\n<p>\n<!------------------------>\n\nThe graph of $y=f(x)$ is <b>symmetric about the origin<\/b> if and\nonly if\n\\[f(-x)=-f(x)\\quad {\\small\\textrm{for all }}x{\\small\\textrm{ in the domain.}}\\]\nHere is an intuitive way to view this symmetry:\n\n<\/p>\n\n\n\n<p>\nIf you start at a point on the curve, draw a line segment through that\npoint and the origin, and extend it an equal distance past the origin,\nyou arrive at another point on the curve.\n\n\n\n<!------------------------>\n\n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p>\nOf the functions in the example, \n<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li> $f(x)=x^2$ is even.\n\t<br><br>\n\t<\/li><li> $f(x)=\\sin x$ is odd.\n\t<br><br>\n\t<\/li><li> The others are neither even nor odd.\n<\/li><\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">Transformations of Functions<\/h4>\n\n\n\n<p>\n\nWe will examine four classes of transformations, each applied to the\nfunction $f(x)=\\sin x$ in the graphing examples.\n\n<\/p>\n\n\n\n<p> <b>Horizontal translation:<\/b> $g(x)=f(x+c)$.<br> The graph is translated $c$ units to the <i>left<\/i> if $c &gt; 0$ and $c$ units to the <i>right<\/i> if $c &lt; 0$.<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"326\" height=\"195\" src=\"https:\/\/i0.wp.com\/104.42.120.246.xip.io\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/trans_sin1.gif?resize=326%2C195\" alt=\"\" class=\"wp-image-410\"\/><\/figure>\n\n\n\n<p> <b>Vertical translation:<\/b> $g(x)=f(x)+k$.<br> The graph is translated $k$ units <i>upward<\/i> if $k &gt; 0$ and $k$ units <i>downward<\/i> if $k &lt; 0$. <\/p>\n\n\n\n<figure class=\"wp-block-image\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"329\" height=\"194\" src=\"https:\/\/i0.wp.com\/104.42.120.246.xip.io\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/trans_sin2.gif?resize=329%2C194\" alt=\"\" class=\"wp-image-411\"\/><\/figure>\n\n\n\n<p> <b>Change of amplitude:<\/b> $g(x)=Af(x)$.<br> The amplitude of the graph is increased by a factor of $A$ if $|A| &gt; 1$ and decreased by a factor of $A$ if $|A| &lt; 1$.  In addition, if $A &lt; 0$ the graph is inverted. <\/p>\n\n\n\n<figure class=\"wp-block-image\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"324\" height=\"195\" src=\"https:\/\/i0.wp.com\/104.42.120.246.xip.io\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/trans_sin3.gif?resize=324%2C195\" alt=\"\" class=\"wp-image-412\"\/><\/figure>\n\n\n\n<p> <b>Change of scale:<\/b>  $g(x)=f(ax)$.<br> The graph is &#8220;compressed&#8221; if $|a| &gt; 1$ and &#8220;stretched out&#8221; if $|a| &lt; 1$.  In addition, if $a &lt; 0$ the graph is reflected about the $y$-axis.<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" width=\"326\" height=\"199\" src=\"https:\/\/i0.wp.com\/104.42.120.246.xip.io\/calculus-tutorials\/wp-content\/uploads\/sites\/3\/2019\/08\/trans_sin4.gif?resize=326%2C199\" alt=\"\" class=\"wp-image-413\"\/><\/figure>\n\n\n\n<center>\n\n<hr color=\"blue\">\n\n<p>\n<\/p><h3>Key Concepts<\/h3>\n<p>\n<\/p><\/center>\n\n\n\n<b>function<\/b>\n\n\n\n<p> F: $A\\longrightarrow B$ is a relation that assigns to each $x \n\\in A$ a unique $y \\in B$. We write $y = f(x)$ and call $y$ the <\/p>\n\n\n\n<b>value of $f$\nat $x$<\/b>\n\n\n\n<p> or the <\/p>\n\n\n\n<b>image of $x$ under $f$<\/b>\n\n\n\n<p>. We also say that $f$ <\/p>\n\n\n\n<b>maps<\/b>\n\n\n\n<p> $x$ to $y$.\n\n<\/p>\n\n\n\n<p>\nThe set $A$ is called the <b>domain<\/b> of $f$. The set of all possible values of \n$f(x)$ in $B$ is called the <b>range<\/b> of $f$.\n\n<\/p>\n\n\n\n<p>\nEach of these transformations takes a function $f$ and produces a new function $g$:\n\n<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li> <b>Horizontal translation:<\/b> $g(x) = f(x+c)$.<br>\n\t\tThe graph is translated $c$ units to the <i>left<\/i> if $c &gt; 0$ and $c$ units to \n\t\tthe <i>right<\/i> if $c &lt; 0$.\n\n\t<br><br>\n\t<\/li><li> <b>Vertical translation:<\/b> $g(x) = f(x)+k$.<br>\n\t\tThe graph is translated $k$ units <i>upward<\/i> if $k &gt; 0$ and $k$ units \n\t\t<i>downward<\/i> if $k &lt; 0$.\n\n\t<br><br>\n\t<\/li><li> <b>Change of amplitude:<\/b> $g(x) = Af(x)$.<br>\n\t\tThe amplitude of the graph is increased by a factor of $A$ if $|A| &gt; 1$ and \n\t\tdecreased by a factor of $A$ if $|A| &lt; 1$. In addition, if $A &lt; 0$ the graph is \n\t\tinverted.\n\n\t<br><br>\n\t<\/li><li> <b>Change of scale:<\/b> $g(x) = f(ax)$.<br>\n\t\tThe graph is &#8220;compressed&#8221; if $|a| &gt; 1$ and &#8220;stretched out&#8221; if $|a| &lt; 1$. \n\t\tIn addition, if $a &lt; 0$ the graph is reflected about the $y$-axis.\n\n<!------------------------>\n\n\n<br>\n\n<hr>\n\n<p>\n\n[<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ0710\/\">I&#8217;m ready to take the quiz.<\/a>]\n[<a href=\"#top\">I need to review more.<\/a>]<br>\n\n\n<\/p><p>\n\n\n\n\n\n<\/p>\n\n\n\n<p><\/p>\n<\/li><\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Functions and Transformations of Functions &#8211; HMC Calculus Tutorial We will review some of the important concepts dealing with functions and transformations of functions. Most likely you have encountered each of these ideas previously, but here we will tie the concepts together. Definition of a Function Let $A$ and $B$ be sets. A function $F:A\\to&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":55,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-154","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/154","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/comments?post=154"}],"version-history":[{"count":9,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/154\/revisions"}],"predecessor-version":[{"id":1073,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/154\/revisions\/1073"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/55"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=154"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=154"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}