{"id":41,"date":"2025-02-13T22:43:05","date_gmt":"2025-02-13T22:43:05","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/domain-and-range\/"},"modified":"2026-01-27T17:26:51","modified_gmt":"2026-01-27T17:26:51","slug":"domain-and-range","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/domain-and-range\/","title":{"raw":"Domain and Range: Learn It 1","rendered":"Domain and Range: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find the domain of a function by looking at its equation<\/li>\r\n \t<li>Find the domain of a function by looking at its graph<\/li>\r\n \t<li>Sketch graphs of piecewise functions<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<h2>Domain and Range<\/h2>\r\nNow that we understand what functions are, let's talk more about two important concepts related to them: the domain and the range.\r\n\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>Domain and Range<\/h3>\r\n<strong>Domain<\/strong>: The domain of a function is the set of all possible input values. These are the values that you can put into the function.\r\n<ul>\r\n \t<li>Note that values in the domain are also known as input values, or values of the <strong>independent variable<\/strong>, and are often labeled with the lowercase letter [latex]x[\/latex].<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\n<strong>Range<\/strong>: The range of a function is the set of all possible output values. These are the values that come out of the function.\r\n<ul>\r\n \t<li>Values in the range are also known as output values, or values of the <strong>dependent variable<\/strong>, and are often labeled with the lowercase letter [latex]y[\/latex].<\/li>\r\n<\/ul>\r\n<\/section>We can write the domain and range in <strong>interval notation<\/strong>, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [latex][[\/latex] when the set includes the endpoint and a parenthesis [latex] ([\/latex] to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has [latex]$100[\/latex] to spend, he or she would need to express the interval that is more than [latex]0[\/latex] and less than or equal to [latex]100[\/latex] and write [latex]\\left(0,\\text{ }100\\right][\/latex].\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">Before we begin, let us review the conventions of interval notation:\r\n<ul>\r\n \t<li>The smallest term from the interval is written first.<\/li>\r\n \t<li>The largest term in the interval is written second, following a comma.<\/li>\r\n \t<li>Parentheses, ( or ), are used to signify that an endpoint is not included, called exclusive.<\/li>\r\n \t<li>Brackets, [ or ], are used to indicate that an endpoint is included, called inclusive.<\/li>\r\n<\/ul>\r\n<\/section>Understanding the domain and range helps us to see the full scope of a function and how it operates over different values.\r\n\r\n<section class=\"textbox example\">Consider the relation where the input is a family member's name and the output is their age:\r\n<table>\r\n<thead>\r\n<tr style=\"height: 30px;\">\r\n<th style=\"height: 30px; text-align: center;\" scope=\"row\">Family Member's Name (Input)<\/th>\r\n<th style=\"height: 30px; text-align: center;\">Family Member's Age (Output)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: center;\" scope=\"row\">Nellie<\/td>\r\n<td style=\"height: 15px; text-align: center;\">[latex]13[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: center;\" scope=\"row\">Marcos<\/td>\r\n<td style=\"height: 15px; text-align: center;\">[latex]11[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.1719px;\">\r\n<td style=\"height: 15.1719px; text-align: center;\" scope=\"row\">Esther<\/td>\r\n<td style=\"height: 15.1719px; text-align: center;\">[latex]46[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: center;\" scope=\"row\">Samuel<\/td>\r\n<td style=\"height: 15px; text-align: center;\">[latex]47[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: center;\" scope=\"row\">Nina<\/td>\r\n<td style=\"height: 15px; text-align: center;\">[latex]47[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: center;\" scope=\"row\">Paul<\/td>\r\n<td style=\"height: 15px; text-align: center;\">[latex]47[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: center;\" scope=\"row\">Katrina<\/td>\r\n<td style=\"height: 15px; text-align: center;\">[latex]21[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: center;\" scope=\"row\">Andrew<\/td>\r\n<td style=\"height: 15px; text-align: center;\">[latex]16[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: center;\" scope=\"row\">Maria<\/td>\r\n<td style=\"height: 15px; text-align: center;\">[latex]13[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px; text-align: center;\" scope=\"row\">Ana<\/td>\r\n<td style=\"height: 15px; text-align: center;\">[latex]81[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<strong>Domain<\/strong>: The domain is the set of all family members\u2019 names: <span class=\"math math-inline\"><span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord text\"><span class=\"mord\">Nellie,\u00a0Marcos,\u00a0Esther,\u00a0Samuel,\u00a0Nina,\u00a0Paul,\u00a0Katrina,\u00a0Andrew,\u00a0Maria,\u00a0Ana<\/span><\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span><\/span>\r\n\r\n<strong>Range<\/strong>: The range is the set of all family members\u2019 ages: [latex]\\{13,11,46,47,21,16,81\\}[\/latex]\r\n\r\n<\/section>Relations can be written as ordered pairs of numbers [latex](x,y)[\/latex] or as numbers in a table of values the columns of which each contain inputs or outputs. By examining the inputs ([latex]x[\/latex]-coordinates) and outputs ([latex]y[\/latex]-coordinates), you can determine whether or not the relation is a function. Remember, in a function, each input corresponds to only one output. That is, each [latex]x[\/latex]<em>\u00a0<\/em>value corresponds to exactly one [latex]y[\/latex] value.\r\n\r\n<section class=\"textbox example\">Find the domain of the following function:<center>[latex]\\left\\{\\left(2,\\text{ }10\\right),\\left(3,\\text{ }10\\right),\\left(4,\\text{ }20\\right),\\left(5,\\text{ }30\\right),\\left(6,\\text{ }40\\right)\\right\\}[\/latex]<\/center>[reveal-answer q=\"202869\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"202869\"]First identify the input values. The input value is the first coordinate in an [pb_glossary id=\"3360\"]ordered pair[\/pb_glossary]. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.\r\n<p style=\"text-align: center;\">[latex]\\left\\{2,3,4,5,6\\right\\}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]317866[\/ohm_question]<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]317867[\/ohm_question]<\/section>\r\n<div class=\"\u201ctextbox\u201d\"><\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find the domain of a function by looking at its equation<\/li>\n<li>Find the domain of a function by looking at its graph<\/li>\n<li>Sketch graphs of piecewise functions<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<h2>Domain and Range<\/h2>\n<p>Now that we understand what functions are, let&#8217;s talk more about two important concepts related to them: the domain and the range.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>Domain and Range<\/h3>\n<p><strong>Domain<\/strong>: The domain of a function is the set of all possible input values. These are the values that you can put into the function.<\/p>\n<ul>\n<li>Note that values in the domain are also known as input values, or values of the <strong>independent variable<\/strong>, and are often labeled with the lowercase letter [latex]x[\/latex].<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><strong>Range<\/strong>: The range of a function is the set of all possible output values. These are the values that come out of the function.<\/p>\n<ul>\n<li>Values in the range are also known as output values, or values of the <strong>dependent variable<\/strong>, and are often labeled with the lowercase letter [latex]y[\/latex].<\/li>\n<\/ul>\n<\/section>\n<p>We can write the domain and range in <strong>interval notation<\/strong>, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [latex][[\/latex] when the set includes the endpoint and a parenthesis [latex]([\/latex] to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has [latex]$100[\/latex] to spend, he or she would need to express the interval that is more than [latex]0[\/latex] and less than or equal to [latex]100[\/latex] and write [latex]\\left(0,\\text{ }100\\right][\/latex].<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">Before we begin, let us review the conventions of interval notation:<\/p>\n<ul>\n<li>The smallest term from the interval is written first.<\/li>\n<li>The largest term in the interval is written second, following a comma.<\/li>\n<li>Parentheses, ( or ), are used to signify that an endpoint is not included, called exclusive.<\/li>\n<li>Brackets, [ or ], are used to indicate that an endpoint is included, called inclusive.<\/li>\n<\/ul>\n<\/section>\n<p>Understanding the domain and range helps us to see the full scope of a function and how it operates over different values.<\/p>\n<section class=\"textbox example\">Consider the relation where the input is a family member&#8217;s name and the output is their age:<\/p>\n<table>\n<thead>\n<tr style=\"height: 30px;\">\n<th style=\"height: 30px; text-align: center;\" scope=\"row\">Family Member&#8217;s Name (Input)<\/th>\n<th style=\"height: 30px; text-align: center;\">Family Member&#8217;s Age (Output)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: center;\" scope=\"row\">Nellie<\/td>\n<td style=\"height: 15px; text-align: center;\">[latex]13[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: center;\" scope=\"row\">Marcos<\/td>\n<td style=\"height: 15px; text-align: center;\">[latex]11[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15.1719px;\">\n<td style=\"height: 15.1719px; text-align: center;\" scope=\"row\">Esther<\/td>\n<td style=\"height: 15.1719px; text-align: center;\">[latex]46[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: center;\" scope=\"row\">Samuel<\/td>\n<td style=\"height: 15px; text-align: center;\">[latex]47[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: center;\" scope=\"row\">Nina<\/td>\n<td style=\"height: 15px; text-align: center;\">[latex]47[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: center;\" scope=\"row\">Paul<\/td>\n<td style=\"height: 15px; text-align: center;\">[latex]47[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: center;\" scope=\"row\">Katrina<\/td>\n<td style=\"height: 15px; text-align: center;\">[latex]21[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: center;\" scope=\"row\">Andrew<\/td>\n<td style=\"height: 15px; text-align: center;\">[latex]16[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: center;\" scope=\"row\">Maria<\/td>\n<td style=\"height: 15px; text-align: center;\">[latex]13[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px; text-align: center;\" scope=\"row\">Ana<\/td>\n<td style=\"height: 15px; text-align: center;\">[latex]81[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Domain<\/strong>: The domain is the set of all family members\u2019 names: <span class=\"math math-inline\"><span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord text\"><span class=\"mord\">Nellie,\u00a0Marcos,\u00a0Esther,\u00a0Samuel,\u00a0Nina,\u00a0Paul,\u00a0Katrina,\u00a0Andrew,\u00a0Maria,\u00a0Ana<\/span><\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span><\/span><\/p>\n<p><strong>Range<\/strong>: The range is the set of all family members\u2019 ages: [latex]\\{13,11,46,47,21,16,81\\}[\/latex]<\/p>\n<\/section>\n<p>Relations can be written as ordered pairs of numbers [latex](x,y)[\/latex] or as numbers in a table of values the columns of which each contain inputs or outputs. By examining the inputs ([latex]x[\/latex]-coordinates) and outputs ([latex]y[\/latex]-coordinates), you can determine whether or not the relation is a function. Remember, in a function, each input corresponds to only one output. That is, each [latex]x[\/latex]<em>\u00a0<\/em>value corresponds to exactly one [latex]y[\/latex] value.<\/p>\n<section class=\"textbox example\">Find the domain of the following function:<\/p>\n<div style=\"text-align: center;\">[latex]\\left\\{\\left(2,\\text{ }10\\right),\\left(3,\\text{ }10\\right),\\left(4,\\text{ }20\\right),\\left(5,\\text{ }30\\right),\\left(6,\\text{ }40\\right)\\right\\}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q202869\">Show Solution<\/button><\/p>\n<div id=\"q202869\" class=\"hidden-answer\" style=\"display: none\">First identify the input values. The input value is the first coordinate in an <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_41_3360\">ordered pair<\/a>. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.<\/p>\n<p style=\"text-align: center;\">[latex]\\left\\{2,3,4,5,6\\right\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm317866\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317866&theme=lumen&iframe_resize_id=ohm317866&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm317867\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317867&theme=lumen&iframe_resize_id=ohm317867&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<div class=\"\u201ctextbox\u201d\"><\/div>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_41_3360\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_41_3360\"><div tabindex=\"-1\"><p>a point written in the form (x,y)<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":6,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"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":36,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/41"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":14,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/41\/revisions"}],"predecessor-version":[{"id":5435,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/41\/revisions\/5435"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/36"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/41\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=41"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=41"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=41"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=41"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}