{"id":586,"date":"2025-10-09T15:23:34","date_gmt":"2025-10-09T15:23:34","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=586"},"modified":"2025-10-23T13:10:56","modified_gmt":"2025-10-23T13:10:56","slug":"functions-of-several-variables-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/functions-of-several-variables-learn-it-1\/","title":{"raw":"Functions of Several Variables: Learn It 1","rendered":"Functions of Several Variables: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Recognize a function of two variables and identify its domain and range<\/li>\r\n \t<li>Sketch a graph of a function of two variables<\/li>\r\n \t<li>Analyze functions by looking at their level curves and traces<\/li>\r\n \t<li>Identify functions of three or more variables and understand their level surfaces<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Functions of Two Variables<\/h2>\r\n<p class=\"whitespace-normal break-words\">Our first step is to explain what a function of more than one variable is, starting with functions of two independent variables. This includes identifying the domain and range of such functions and learning how to graph them. We also examine ways to relate graphs of functions in three dimensions to graphs of more familiar planar functions.<\/p>\r\n<p class=\"whitespace-normal break-words\">The definition of a function of two variables is very similar to the definition for a function of one variable. The main difference is that instead of mapping values of one variable to values of another variable, we map ordered pairs of variables to another variable.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>function of two variables<\/h3>\r\n<p class=\"whitespace-normal break-words\">A function of two variables [latex]z=f(x, y)[\/latex] maps each ordered pair [latex](x, y)[\/latex] in a subset [latex]{\\bf{D}}[\/latex] of the real plane [latex]\\mathbb{R}^{2}[\/latex] to a unique real number [latex]z[\/latex].<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">The set [latex]{\\bf{D}}[\/latex] is called the <strong>domain<\/strong> of the function<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The <strong>range<\/strong> of [latex]f[\/latex] is the set of all real numbers [latex]z[\/latex] that has at least one ordered pair [latex](x, y)\\in{\\bf{D}}[\/latex] such that [latex]f(x, y)=z[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>[caption id=\"attachment_944\" align=\"aligncenter\" width=\"489\"]<img class=\"size-full wp-image-944\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/27163914\/4-1-1.jpeg\" alt=\"A bulbous shape is marked domain and it contains the point (x, y). From this point, there is an arrow marked f that points to a point z on a straight line marked range.\" width=\"489\" height=\"210\" \/> Figure 1.\u00a0The domain of a function of two variables consists of ordered pairs [latex](x,y)[\/latex].[\/caption]\r\n<p class=\"whitespace-normal break-words\">Determining the domain of a function of two variables involves taking into account any domain restrictions that may exist.<\/p>\r\n\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<p class=\"whitespace-normal break-words\">Look for restrictions similar to those in single-variable calculus:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Square roots require non-negative expressions under the radical<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Denominators cannot equal zero<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Logarithms require positive arguments<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">The difference is that these restrictions now define regions in the [latex]xy[\/latex]-plane rather than intervals on the number line.<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Find the domain and range of each of the following functions:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]f\\,(x,\\ y)=3x+5y+2[\/latex]<\/li>\r\n \t<li>[latex]g\\,(x,\\ y)=\\sqrt{9-x^{2}-y^{2}}[\/latex]<\/li>\r\n<\/ol>\r\n<div id=\"fs-id1167793361764\" class=\"exercise\">[reveal-answer q=\"fs-id1167794055154\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794055154\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>This is an example of a linear function in two variables. There are no values or combinations of [latex]x[\/latex] and [latex]y[\/latex] that cause [latex]f\\,(x,\\ y)[\/latex] to be undefined, so the domain of [latex]f[\/latex] is [latex]\\mathbb{R}^{2}[\/latex]. To determine the range, first pick a value for [latex]z[\/latex]. We need to find a solution to the equation [latex]f\\,(x,\\ y)=z[\/latex], or [latex]3x-5y+2=z[\/latex]. One such solution can be obtained by first setting [latex]y=0[\/latex], which yields the equation [latex]3x+2=z[\/latex]. The solution to this equation is [latex]x=\\frac{z-2}{3}[\/latex], which gives the ordered pair [latex](\\frac{z-2}{3},\\ 0)[\/latex] as a solution to the equation [latex]f\\,(x,\\ y)=z[\/latex] for any value of [latex]z[\/latex]. Therefore, the range of the function is all real numbers, or [latex]\\mathbb{R}[\/latex].<\/li>\r\n \t<li>For the function [latex]g\\,(x,\\ y)[\/latex] to have a real value, the quantity under the square root must be nonnegative:\r\n<div style=\"text-align: center;\">[latex]9-x^{2}-y^{2}\\,\\geq\\,0.[\/latex]<\/div>\r\n&nbsp;\r\n\r\nThis inequality can be written in the form\r\n<div style=\"text-align: center;\">[latex]x^{2}+y^{2}\\,\\leq\\,9.[\/latex]<\/div>\r\n&nbsp;\r\n\r\nTherefore, the domain of [latex]g\\,(x,\\ y)[\/latex] is [latex]{\\{(x,\\ y)\\in\\mathbb{R}^{2}\\,|\\,x^{2}+y^{2}\\,\\leq\\,9\\}}[\/latex]. The graph of this set of points can be described as a disk of radius [latex]3[\/latex] centered at the origin. The domain includes the boundary circle as shown in the following graph.\r\n<div>[caption id=\"attachment_946\" align=\"aligncenter\" width=\"417\"]<img class=\"size-full wp-image-946\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/27164019\/4-1-2.jpeg\" alt=\"A circle of radius three with center at the origin. The equation x2 + y2 = 9 is given.\" width=\"417\" height=\"422\" \/> Figure 2.\u00a0The domain of the function [latex]\\small{g(x,y)=\\sqrt{9-x^{2}-y^{2}}}[\/latex] is a closed disk of radius [latex]\\small{3}[\/latex].[\/caption]<\/div>\r\nTo determine the range of [latex]g\\,(x,\\ y)=\\sqrt{9-x^{2}-y^{2}}[\/latex] we start with a point [latex](x_0,\\ y_0)[\/latex] on the boundary of the domain, which is defined by the relation [latex]x^{2}+y^{2}=9[\/latex]. It follows that [latex]x^2_0+y^2_0=9[\/latex] and\r\n<div style=\"text-align: center;\">[latex]g\\,(x_0,\\ y_0)=\\sqrt{9-x^2_0-y^2_0}=\\sqrt{9-(x^2_0+y^2_0)}=\\sqrt{9-9}=0[\/latex].<\/div>\r\n&nbsp;\r\n\r\nIf [latex]x^2_0+y^2_0=0[\/latex] (in other words, [latex]x_0=y_0=0[\/latex]), then\r\n<div style=\"text-align: center;\">[latex]g\\,(x_0,\\ y_0)=\\sqrt{9-x^2_0-y^2_0}=\\sqrt{9-(x^2_0+y^2_0)}=\\sqrt{9-0}=3[\/latex]<\/div>\r\n&nbsp;\r\n\r\nThis is the maximum value of the function. Given any value [latex]c[\/latex] between 0 and 3, we can find an entire set of points inside the domain of [latex]g[\/latex] such that [latex]g\\,(x,\\ y)=c[\/latex]:\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc} \\hfill {\\sqrt{9-x^{2}-y^{2}}} &amp; =\\hfill &amp; {c} \\hfill \\\\ \\hfill {9-x^{2}-y^{2}} &amp; =\\hfill &amp; {c^{2}}\\hfill \\\\ \\hfill {x^{2}+y^{2}} &amp; =\\hfill &amp; {9-c^{2}.}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n\r\nSince [latex]9-c^{2}\\,&gt;\\,0[\/latex], this describes a circle of radius [latex]\\sqrt{9-c^{2}}[\/latex] centered at the origin. Any point on this circle satisfies the equation [latex]g\\,(x,\\ y)=c[\/latex]. Therefore, the range of this function can be written in interval notation as [latex][0,\\ 3][\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Recognize a function of two variables and identify its domain and range<\/li>\n<li>Sketch a graph of a function of two variables<\/li>\n<li>Analyze functions by looking at their level curves and traces<\/li>\n<li>Identify functions of three or more variables and understand their level surfaces<\/li>\n<\/ul>\n<\/section>\n<h2>Functions of Two Variables<\/h2>\n<p class=\"whitespace-normal break-words\">Our first step is to explain what a function of more than one variable is, starting with functions of two independent variables. This includes identifying the domain and range of such functions and learning how to graph them. We also examine ways to relate graphs of functions in three dimensions to graphs of more familiar planar functions.<\/p>\n<p class=\"whitespace-normal break-words\">The definition of a function of two variables is very similar to the definition for a function of one variable. The main difference is that instead of mapping values of one variable to values of another variable, we map ordered pairs of variables to another variable.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>function of two variables<\/h3>\n<p class=\"whitespace-normal break-words\">A function of two variables [latex]z=f(x, y)[\/latex] maps each ordered pair [latex](x, y)[\/latex] in a subset [latex]{\\bf{D}}[\/latex] of the real plane [latex]\\mathbb{R}^{2}[\/latex] to a unique real number [latex]z[\/latex].<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">The set [latex]{\\bf{D}}[\/latex] is called the <strong>domain<\/strong> of the function<\/li>\n<li class=\"whitespace-normal break-words\">The <strong>range<\/strong> of [latex]f[\/latex] is the set of all real numbers [latex]z[\/latex] that has at least one ordered pair [latex](x, y)\\in{\\bf{D}}[\/latex] such that [latex]f(x, y)=z[\/latex]<\/li>\n<\/ul>\n<\/section>\n<figure id=\"attachment_944\" aria-describedby=\"caption-attachment-944\" style=\"width: 489px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-944\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/27163914\/4-1-1.jpeg\" alt=\"A bulbous shape is marked domain and it contains the point (x, y). From this point, there is an arrow marked f that points to a point z on a straight line marked range.\" width=\"489\" height=\"210\" \/><figcaption id=\"caption-attachment-944\" class=\"wp-caption-text\">Figure 1.\u00a0The domain of a function of two variables consists of ordered pairs [latex](x,y)[\/latex].<\/figcaption><\/figure>\n<p class=\"whitespace-normal break-words\">Determining the domain of a function of two variables involves taking into account any domain restrictions that may exist.<\/p>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<p class=\"whitespace-normal break-words\">Look for restrictions similar to those in single-variable calculus:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Square roots require non-negative expressions under the radical<\/li>\n<li class=\"whitespace-normal break-words\">Denominators cannot equal zero<\/li>\n<li class=\"whitespace-normal break-words\">Logarithms require positive arguments<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">The difference is that these restrictions now define regions in the [latex]xy[\/latex]-plane rather than intervals on the number line.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Find the domain and range of each of the following functions:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]f\\,(x,\\ y)=3x+5y+2[\/latex]<\/li>\n<li>[latex]g\\,(x,\\ y)=\\sqrt{9-x^{2}-y^{2}}[\/latex]<\/li>\n<\/ol>\n<div id=\"fs-id1167793361764\" class=\"exercise\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167794055154\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167794055154\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>This is an example of a linear function in two variables. There are no values or combinations of [latex]x[\/latex] and [latex]y[\/latex] that cause [latex]f\\,(x,\\ y)[\/latex] to be undefined, so the domain of [latex]f[\/latex] is [latex]\\mathbb{R}^{2}[\/latex]. To determine the range, first pick a value for [latex]z[\/latex]. We need to find a solution to the equation [latex]f\\,(x,\\ y)=z[\/latex], or [latex]3x-5y+2=z[\/latex]. One such solution can be obtained by first setting [latex]y=0[\/latex], which yields the equation [latex]3x+2=z[\/latex]. The solution to this equation is [latex]x=\\frac{z-2}{3}[\/latex], which gives the ordered pair [latex](\\frac{z-2}{3},\\ 0)[\/latex] as a solution to the equation [latex]f\\,(x,\\ y)=z[\/latex] for any value of [latex]z[\/latex]. Therefore, the range of the function is all real numbers, or [latex]\\mathbb{R}[\/latex].<\/li>\n<li>For the function [latex]g\\,(x,\\ y)[\/latex] to have a real value, the quantity under the square root must be nonnegative:\n<div style=\"text-align: center;\">[latex]9-x^{2}-y^{2}\\,\\geq\\,0.[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>This inequality can be written in the form<\/p>\n<div style=\"text-align: center;\">[latex]x^{2}+y^{2}\\,\\leq\\,9.[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>Therefore, the domain of [latex]g\\,(x,\\ y)[\/latex] is [latex]{\\{(x,\\ y)\\in\\mathbb{R}^{2}\\,|\\,x^{2}+y^{2}\\,\\leq\\,9\\}}[\/latex]. The graph of this set of points can be described as a disk of radius [latex]3[\/latex] centered at the origin. The domain includes the boundary circle as shown in the following graph.<\/p>\n<div>\n<figure id=\"attachment_946\" aria-describedby=\"caption-attachment-946\" style=\"width: 417px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-946\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/27164019\/4-1-2.jpeg\" alt=\"A circle of radius three with center at the origin. The equation x2 + y2 = 9 is given.\" width=\"417\" height=\"422\" \/><figcaption id=\"caption-attachment-946\" class=\"wp-caption-text\">Figure 2.\u00a0The domain of the function [latex]\\small{g(x,y)=\\sqrt{9-x^{2}-y^{2}}}[\/latex] is a closed disk of radius [latex]\\small{3}[\/latex].<\/figcaption><\/figure>\n<\/div>\n<p>To determine the range of [latex]g\\,(x,\\ y)=\\sqrt{9-x^{2}-y^{2}}[\/latex] we start with a point [latex](x_0,\\ y_0)[\/latex] on the boundary of the domain, which is defined by the relation [latex]x^{2}+y^{2}=9[\/latex]. It follows that [latex]x^2_0+y^2_0=9[\/latex] and<\/p>\n<div style=\"text-align: center;\">[latex]g\\,(x_0,\\ y_0)=\\sqrt{9-x^2_0-y^2_0}=\\sqrt{9-(x^2_0+y^2_0)}=\\sqrt{9-9}=0[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p>If [latex]x^2_0+y^2_0=0[\/latex] (in other words, [latex]x_0=y_0=0[\/latex]), then<\/p>\n<div style=\"text-align: center;\">[latex]g\\,(x_0,\\ y_0)=\\sqrt{9-x^2_0-y^2_0}=\\sqrt{9-(x^2_0+y^2_0)}=\\sqrt{9-0}=3[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>This is the maximum value of the function. Given any value [latex]c[\/latex] between 0 and 3, we can find an entire set of points inside the domain of [latex]g[\/latex] such that [latex]g\\,(x,\\ y)=c[\/latex]:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc} \\hfill {\\sqrt{9-x^{2}-y^{2}}} & =\\hfill & {c} \\hfill \\\\ \\hfill {9-x^{2}-y^{2}} & =\\hfill & {c^{2}}\\hfill \\\\ \\hfill {x^{2}+y^{2}} & =\\hfill & {9-c^{2}.}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>Since [latex]9-c^{2}\\,>\\,0[\/latex], this describes a circle of radius [latex]\\sqrt{9-c^{2}}[\/latex] centered at the origin. Any point on this circle satisfies the equation [latex]g\\,(x,\\ y)=c[\/latex]. Therefore, the range of this function can be written in interval notation as [latex][0,\\ 3][\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":5,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-586","chapter","type-chapter","status-publish","hentry"],"part":426,"module-header":"learn_it","content_attributions":[],"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\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/586","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/586\/revisions"}],"predecessor-version":[{"id":1733,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/586\/revisions\/1733"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/426"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/586\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=586"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=586"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=586"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=586"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}