{"id":29059,"date":"2021-06-26T13:00:38","date_gmt":"2021-06-26T13:00:38","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29059"},"modified":"2024-09-22T07:03:33","modified_gmt":"2024-09-22T07:03:33","slug":"domain-range-and-graph-of-algebraic-functions","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/","title":{"rendered":"Domain, Range, and Graph of Algebraic Functions"},"content":{"rendered":"<p><center><\/p>\n<h1>Domain, Range, and Graph of Algebraic Functions<\/h1>\n<p><em><strong>Summary:<\/strong><br \/>\nThis class introduces the concepts of domain, range, and graph of functions, applying them to practical examples of algebraic functions. Graphical and analytical techniques for determining these elements are reviewed.<br \/>\n<\/em><br \/>\n<strong>Learning Objectives:<\/strong><br \/>\nAt the end of this class, the student will be able to <\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Correctly define<\/strong> the domain, range, and graph of a function.<\/li>\n<li><strong>Apply<\/strong> graphical methods to determine the domain and range of algebraic functions.<\/li>\n<li><strong>Build<\/strong> sign tables to analyze the behavior of functions.<\/li>\n<\/ol>\n<p><\/center><\/p>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/zhb8GKlcdA8\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><br \/>\n<\/center><\/p>\n<h2>Definition of Domain, Range, and Graph<\/h2>\n<p style=\"text-align: justify;\">By now, we have conducted a fairly detailed study on linear, quadratic, and similar functions. We also studied curves such as lines, parabolas, ellipses, and hyperbolas, as well as operations with polynomials and algebraic functions in general. With this done, it will now be much easier to delve into some more foundational aspects regarding functions in general, which we will begin reviewing by introducing the concepts of <strong>domain, range, and graph.<\/strong><\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=zhb8GKlcdA8&amp;t=306s\" target=\"_blank\" rel=\"noopener\"><strong>Let <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> be a function<\/strong><\/a> defined between sets <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{matrix}f &amp; : &amp; A &amp; \\longrightarrow &amp; B \\\\ &amp; &amp; x &amp; \\longmapsto &amp; y=f(x)\n\n\\end{matrix}<\/span>\n<p style=\"text-align: justify;\">Sets <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> are referred to as the \u00abinput\u00bb and \u00aboutput\u00bb sets, respectively. From these, the following sets are defined:<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">Dom(f) = \\{x\\in A\\;|\\; (\\exists y \\in B)(y=f(x))\\}<\/span>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">Rec(f) = \\{y\\in B\\;|\\; (\\exists ! x \\in Dom(f))(y=f(x))\\}<\/span>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">Graf(f) = \\{(x,y)\\in A\\times B\\;|\\; x\\in Dom(f) \\wedge y=f(x) \\}<\/span>\n<h2>Example Analysis<\/h2>\n<p style=\"text-align: justify;\">Although everything that can be learned about the concepts of domain, range, and graph is essentially theoretical, understanding comes more from practical examples, which we will now analyze by reviewing the following three cases:<\/p>\n<h3>Calculate the domain, range, and graph of: <span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\sqrt{1-x^2}<\/span><\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=zhb8GKlcdA8&amp;t=560s\" target=\"_blank\" rel=\"noopener\"><strong>Let&#8217;s start this analysis<\/strong><\/a> by writing <span class=\"katex-eq\" data-katex-display=\"false\">y=f(x).<\/span> Doing so gives us the equation<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">y = \\sqrt{1-x^2}<\/span>\n<p style=\"text-align: justify;\">If we square this expression, we quickly arrive at an equation leading to things we already know<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n&amp; y^2 = 1-x^2 \\\\\n\n\\equiv &amp; x^2 + y^2 = 1 \\end{array}<\/span>\n<p style=\"text-align: justify;\">This is the equation of the unit circle.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-DQGthMyBY6g\/YNVVrnVQEfI\/AAAAAAAAFOQ\/6_lf8fRQdDIT9NMqstyLOJ2F7nQM9pc8ACLcBGAsYHQ\/s0\/circulounitario.PNG\" alt=\"Unit Circle and the Domain, Range, and Graph\" class=\"aligncenter lazyload\" width=\"245\" height=\"249\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-DQGthMyBY6g\/YNVVrnVQEfI\/AAAAAAAAFOQ\/6_lf8fRQdDIT9NMqstyLOJ2F7nQM9pc8ACLcBGAsYHQ\/s0\/circulounitario.PNG\" alt=\"Unit Circle and the Domain, Range, and Graph\" class=\"aligncenter lazyload\" width=\"245\" height=\"249\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">However, we must be careful here, because by squaring we have \u00abadded some information\u00bb. Algebraically, two values satisfy the condition of \u00abbeing the square root of\u00bb, but initially, the square root is specified as a function, and functions only admit a single result. We are talking about the principal root. For this reason, the original formulation refers only to the upper part of the circle, rather than the full figure.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-AxSf-9lgnuE\/YNVbSJpd-rI\/AAAAAAAAFOg\/0APXEMWIFpAm8DX9651iD6wcq5bTJwFoQCLcBGAsYHQ\/s0\/circulounitario%2B2.PNG\" alt=\"unit circle and domain, range, and graph\" class=\" aligncenter lazyload\" width=\"401\" height=\"361\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-AxSf-9lgnuE\/YNVbSJpd-rI\/AAAAAAAAFOg\/0APXEMWIFpAm8DX9651iD6wcq5bTJwFoQCLcBGAsYHQ\/s0\/circulounitario%2B2.PNG\" alt=\"unit circle and domain, range, and graph\" class=\" aligncenter lazyload\" width=\"401\" height=\"361\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">From this figure, it is clear that<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">Dom(f) = \\{x\\in\\mathbb{R}\\;|\\; |x|\\leq 1\\} = [-1,1]<\/span>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">Rec(f) = \\{y\\in\\mathbb{R}\\;|\\; 0\\leq y\\leq 1\\} = [0,1]<\/span>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">Graf(f) = \\{(x,y)\\in \\mathbb{R}\\times \\mathbb{R}\\;|\\; x\\in [-1,1] \\wedge y=\\sqrt{1-x^2}\\}<\/span>\n<p style=\"text-align: justify;\">Although I have conducted this analysis from a graphical perspective, it is also possible to do this from a more analytical approach by reviewing the operations involved.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">f(x) = \\color{red}{\\sqrt{{1-x^2}}}<\/span>\n<p style=\"text-align: justify;\">The part <span class=\"katex-eq\" data-katex-display=\"false\">1-x^2<\/span> is well-defined for all reals.<\/p>\n<p style=\"text-align: justify;\">However, the square root only admits values greater than or equal to zero.<\/p>\n<p style=\"text-align: justify;\">From this, we have:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rlrl}\n\nx\\in Dom(f) &amp; \\leftrightarrow &amp; 0 &amp;\\leq 1-x^2 \\\\\n\n{} &amp; \\leftrightarrow &amp; x^2 &amp;\\leq 1 \\\\\n\n&amp; \\leftrightarrow &amp; |x| &amp;\\leq 1 \\\\\n\n&amp; \\leftrightarrow &amp; -1 &amp;\\leq x \\leq 1 \\\\\n\n\\end{array}\n\n<\/span>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\nTherefore:\\; Dom(f) = \\{x\\in \\mathbb{R}\\;|x| \\leq 1\\} = [-1,1]\n<\/span>\n<p style=\"text-align: justify;\">Analytical methods for determining the range are generally much more complicated; simpler cases are solved by finding the inverse function, but before reviewing this topic in detail, it is advisable to first study function composition and other simpler cases to have a solid foundation. Meanwhile, the graphical methods we will soon review will cover much of the difficulties involved in determining the range.<\/p>\n<h3>Analysis for: <span class=\"katex-eq\" data-katex-display=\"false\">g(x) =\\displaystyle \\frac{x^2 - 1}{x^2 + 1}<\/span><\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=zhb8GKlcdA8&amp;t=1049s\" target=\"_blank\" rel=\"noopener\"><strong>One way to quickly<\/strong><\/a> find the domain of the function is by asking for the values of <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> that \u00abspoil the function.\u00bb It is clear that the function only breaks when the denominator equals zero. That is:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n&amp; x^2 + 1 = 0 \\\\\n\n\\equiv &amp; x^2 = -1 \\\\\n\n\\end{array}<\/span>\n<p style=\"text-align: justify;\">As no real number can satisfy this condition, it is clear that<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\color{blue}{Dom(g) = \\mathbb{R}}<\/span>\n<p style=\"text-align: justify;\">Determining the graph is generally the quickest way to determine the range of a function; and to achieve this, <a href=\"https:\/\/toposuranos.com\/algebra-de-polinomios-de-numeros-reales\/\" rel=\"noopener\" target=\"_blank\">polynomial division<\/a> will be a good tool.<\/p>\n<p style=\"text-align: justify;\">By performing polynomial division, we arrive at:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">y= \\displaystyle\\frac{x^2-1}{x^2+1} =<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> <span class=\"katex-eq\" data-katex-display=\"false\">-\\displaystyle\\frac{2}{x^2 + 1}<\/span>\n<p style=\"text-align: justify;\">Thus, we have separated the original function into two simpler parts that we call the \u00abwhole\u00bb and \u00abfractional\u00bb parts. Plotting each of these parts separately is much easier than plotting the original function all at once.<\/p>\n<h3>Analysis for: <span class=\"katex-eq\" data-katex-display=\"false\">h(x) =\\displaystyle \\frac{x - 1}{\\sqrt{x+1}}<\/span><\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=zhb8GKlcdA8&amp;t=1580s\" target=\"_blank\" rel=\"noopener\"><strong>An algebraic analysis<\/strong><\/a> will help to quickly determine the domain of this function. It is enough to notice that it will be well-defined as long as<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rrl}\n\n&amp; 0 &amp; \\lt x + 1 \\\\\n\n\\equiv &amp; -1 &amp; \\lt x \\\\\n\n\\end{array}\n\n<\/span>\n<p style=\"text-align: justify;\">Therefore, it is clear that <span class=\"katex-eq\" data-katex-display=\"false\">Dom(h)=]-1,+\\infty[.<\/span>\n<p style=\"text-align: justify;\">To find the range, it is useful to sketch the graph, and to do this simply, we will use a <strong>sign table.<\/strong> The function <span class=\"katex-eq\" data-katex-display=\"false\">h(x)<\/span> consists of two parts<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">h(x)=\\displaystyle\\frac{\\color{green}{x-1}}{\\color{red}{\\sqrt{x+1}}}<\/span>\n<p style=\"text-align: justify;\">The top part is zero at <span class=\"katex-eq\" data-katex-display=\"false\">x=1<\/span>; the bottom part, besides becoming zero at <span class=\"katex-eq\" data-katex-display=\"false\">x=-1<\/span>, becomes undefined if <span class=\"katex-eq\" data-katex-display=\"false\">x\\lt-1<\/span>. With this information, we can construct the following sign table:<\/p>\n<table>\n<tbody>\n<tr>\n<th style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/th>\n<th style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">-\\infty<\/span><\/th>\n<th style=\"text-align: center;\"><\/th>\n<th style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">-1<\/span><\/th>\n<th style=\"text-align: center;\"><\/th>\n<th style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">+1<\/span><\/th>\n<th style=\"text-align: center;\"><\/th>\n<th style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">+\\infty<\/span><\/th>\n<\/tr>\n<tr>\n<th style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">x-1<\/span><\/th>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">-\\infty <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> - <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">{} - <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> - <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> 0 <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> + <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">{} +\\infty <\/span><\/td>\n<\/tr>\n<tr>\n<th style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{x+1}<\/span><\/th>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> Not\\,Defined  <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> Not\\,Defined <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> 0 <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> + <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">{} + <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> + <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">{} + <\/span><\/td>\n<\/tr>\n<tr>\n<th style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\frac{x-1}{\\sqrt{x+1}}<\/span><\/th>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> Not\\,Defined <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">{}Not\\,Defined <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> -\\infty <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">{} - <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> 0 <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> + <\/span><\/td>\n<td style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">{} +\\infty <\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">With the information displayed in this table, it is now very simple to graph the function.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-mWc6Hza3Wl0\/YNWMYho7pPI\/AAAAAAAAFO4\/0D8zrIeKcc8HY7hlWuvJOWDnYE6Zw--cQCLcBGAsYHQ\/s0\/grafico%2B2.PNG\" alt=\"domain, range, and graph with sign table\" class=\" aligncenter lazyload\" width=\"498\" height=\"310\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-mWc6Hza3Wl0\/YNWMYho7pPI\/AAAAAAAAFO4\/0D8zrIeKcc8HY7hlWuvJOWDnYE6Zw--cQCLcBGAsYHQ\/s0\/grafico%2B2.PNG\" alt=\"domain, range, and graph with sign table\" class=\" aligncenter lazyload\" width=\"498\" height=\"310\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">And with this, determining the domain and range is now a trivial matter:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">Dom(h)=]-1,+\\infty[<\/span>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">Rec(h)=\\mathbb{R}<\/span>\n<h3>Proposed Exercise<\/h3>\n<p style=\"text-align: justify;\">Using the tools we just reviewed, find the domain, range, and graph of the following function<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">F(x) = \\displaystyle\\frac{4x^3 + 6x^2 -2x + 1}{x^2-4}<\/span>\n","protected":false},"excerpt":{"rendered":"<p>Domain, Range, and Graph of Algebraic Functions Summary: This class introduces the concepts of domain, range, and graph of functions, applying them to practical examples of algebraic functions. Graphical and analytical techniques for determining these elements are reviewed. Learning Objectives: At the end of this class, the student will be able to Correctly define the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29058,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":2,"footnotes":""},"categories":[583,567],"tags":[],"class_list":["post-29059","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-and-geometry","category-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Domain, Range, and Graph of Algebraic Functions - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"The domain, range, and graph of a function are fundamental concepts in mathematics. The domain represents the allowed input values, the range the possible results, and the graph visualizes the relationship between them.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Domain, Range, and Graph of Algebraic Functions\" \/>\n<meta property=\"og:description\" content=\"The domain, range, and graph of a function are fundamental concepts in mathematics. The domain represents the allowed input values, the range the possible results, and the graph visualizes the relationship between them.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-06-26T13:00:38+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-09-22T07:03:33+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/graficos-1024x466.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Domain, Range, and Graph of Algebraic Functions\" \/>\n<meta name=\"twitter:description\" content=\"The domain, range, and graph of a function are fundamental concepts in mathematics. The domain represents the allowed input values, the range the possible results, and the graph visualizes the relationship between them.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/graficos.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"6 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/#article\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Domain, Range, and Graph of Algebraic Functions\",\"datePublished\":\"2021-06-26T13:00:38+00:00\",\"dateModified\":\"2024-09-22T07:03:33+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/\"},\"wordCount\":1154,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/graficos.jpg\",\"articleSection\":[\"Algebra and Geometry\",\"Mathematics\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/\",\"url\":\"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/\",\"name\":\"Domain, Range, and Graph of Algebraic Functions - toposuranos.com\/material\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#website\"},\"primaryImageOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/#primaryimage\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/graficos.jpg\",\"datePublished\":\"2021-06-26T13:00:38+00:00\",\"dateModified\":\"2024-09-22T07:03:33+00:00\",\"description\":\"The domain, range, and graph of a function are fundamental concepts in mathematics. The domain represents the allowed input values, the range the possible results, and the graph visualizes the relationship between them.\",\"breadcrumb\":{\"@id\":\"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/#breadcrumb\"},\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/#primaryimage\",\"url\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/graficos.jpg\",\"contentUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/graficos.jpg\",\"width\":1792,\"height\":815},{\"@type\":\"BreadcrumbList\",\"@id\":\"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Portada\",\"item\":\"https:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Domain, Range, and Graph of Algebraic Functions\"}]},{\"@type\":\"WebSite\",\"@id\":\"http:\/\/toposuranos.com\/material\/#website\",\"url\":\"http:\/\/toposuranos.com\/material\/\",\"name\":\"toposuranos.com\/material\",\"description\":\"\",\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"http:\/\/toposuranos.com\/material\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"es\"},{\"@type\":\"Organization\",\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\",\"name\":\"toposuranos.com\/material\",\"url\":\"http:\/\/toposuranos.com\/material\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/\",\"url\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png\",\"contentUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png\",\"width\":2400,\"height\":2059,\"caption\":\"toposuranos.com\/material\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.facebook.com\/groups\/toposuranos\",\"https:\/\/x.com\/topuranos\",\"https:\/\/www.youtube.com\/channel\/UC16yDm12cPcrwsE0fAM7X1g\",\"https:\/\/www.linkedin.com\/company\/69429190\"]},{\"@type\":\"Person\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\",\"name\":\"giorgio.reveco\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/image\/\",\"url\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg\",\"contentUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg\",\"caption\":\"giorgio.reveco\"},\"description\":\"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.\",\"sameAs\":[\"http:\/\/toposuranos.com\/material\"],\"url\":\"http:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Domain, Range, and Graph of Algebraic Functions - toposuranos.com\/material","description":"The domain, range, and graph of a function are fundamental concepts in mathematics. The domain represents the allowed input values, the range the possible results, and the graph visualizes the relationship between them.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/","og_locale":"es_ES","og_type":"article","og_title":"Domain, Range, and Graph of Algebraic Functions","og_description":"The domain, range, and graph of a function are fundamental concepts in mathematics. The domain represents the allowed input values, the range the possible results, and the graph visualizes the relationship between them.","og_url":"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/","og_site_name":"toposuranos.com\/material","article_publisher":"https:\/\/www.facebook.com\/groups\/toposuranos","article_published_time":"2021-06-26T13:00:38+00:00","article_modified_time":"2024-09-22T07:03:33+00:00","og_image":[{"url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/graficos-1024x466.jpg","type":"","width":"","height":""}],"author":"giorgio.reveco","twitter_card":"summary_large_image","twitter_title":"Domain, Range, and Graph of Algebraic Functions","twitter_description":"The domain, range, and graph of a function are fundamental concepts in mathematics. The domain represents the allowed input values, the range the possible results, and the graph visualizes the relationship between them.","twitter_image":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/graficos.jpg","twitter_creator":"@topuranos","twitter_site":"@topuranos","twitter_misc":{"Escrito por":"giorgio.reveco","Tiempo de lectura":"6 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/#article","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/"},"author":{"name":"giorgio.reveco","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1"},"headline":"Domain, Range, and Graph of Algebraic Functions","datePublished":"2021-06-26T13:00:38+00:00","dateModified":"2024-09-22T07:03:33+00:00","mainEntityOfPage":{"@id":"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/"},"wordCount":1154,"commentCount":0,"publisher":{"@id":"http:\/\/toposuranos.com\/material\/#organization"},"image":{"@id":"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/#primaryimage"},"thumbnailUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/graficos.jpg","articleSection":["Algebra and Geometry","Mathematics"],"inLanguage":"es","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/#respond"]}]},{"@type":"WebPage","@id":"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/","url":"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/","name":"Domain, Range, and Graph of Algebraic Functions - toposuranos.com\/material","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/#website"},"primaryImageOfPage":{"@id":"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/#primaryimage"},"image":{"@id":"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/#primaryimage"},"thumbnailUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/graficos.jpg","datePublished":"2021-06-26T13:00:38+00:00","dateModified":"2024-09-22T07:03:33+00:00","description":"The domain, range, and graph of a function are fundamental concepts in mathematics. The domain represents the allowed input values, the range the possible results, and the graph visualizes the relationship between them.","breadcrumb":{"@id":"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/#breadcrumb"},"inLanguage":"es","potentialAction":[{"@type":"ReadAction","target":["http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/"]}]},{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/#primaryimage","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/graficos.jpg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/graficos.jpg","width":1792,"height":815},{"@type":"BreadcrumbList","@id":"http:\/\/toposuranos.com\/material\/en\/domain-range-and-graph-of-algebraic-functions\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Portada","item":"https:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/"},{"@type":"ListItem","position":2,"name":"Domain, Range, and Graph of Algebraic Functions"}]},{"@type":"WebSite","@id":"http:\/\/toposuranos.com\/material\/#website","url":"http:\/\/toposuranos.com\/material\/","name":"toposuranos.com\/material","description":"","publisher":{"@id":"http:\/\/toposuranos.com\/material\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"http:\/\/toposuranos.com\/material\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"es"},{"@type":"Organization","@id":"http:\/\/toposuranos.com\/material\/#organization","name":"toposuranos.com\/material","url":"http:\/\/toposuranos.com\/material\/","logo":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","width":2400,"height":2059,"caption":"toposuranos.com\/material"},"image":{"@id":"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/groups\/toposuranos","https:\/\/x.com\/topuranos","https:\/\/www.youtube.com\/channel\/UC16yDm12cPcrwsE0fAM7X1g","https:\/\/www.linkedin.com\/company\/69429190"]},{"@type":"Person","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1","name":"giorgio.reveco","image":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/image\/","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","caption":"giorgio.reveco"},"description":"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.","sameAs":["http:\/\/toposuranos.com\/material"],"url":"http:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/"}]}},"_links":{"self":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/29059","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/comments?post=29059"}],"version-history":[{"count":0,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/29059\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media\/29058"}],"wp:attachment":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media?parent=29059"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/categories?post=29059"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/tags?post=29059"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}