{"id":473,"date":"2025-07-10T16:42:22","date_gmt":"2025-07-10T16:42:22","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=473"},"modified":"2026-01-06T22:00:28","modified_gmt":"2026-01-06T22:00:28","slug":"introduction-to-functions-background-youll-need","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/introduction-to-functions-background-youll-need\/","title":{"raw":"Introduction to Functions: Background You'll Need 4","rendered":"Introduction to Functions: Background You&#8217;ll Need 4"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Use interval notation<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"font-600 text-xl font-bold\">Set Notation: Inequality, Set-Builder, and Interval Notations<\/h2>\r\nIn mathematics, we often need to describe sets of numbers that satisfy certain conditions. There are several ways to represent these sets, each with its own advantages and uses in different contexts. This page introduces three common notations for describing sets of numbers: inequality notation, set-builder notation, and interval notation.\r\n\r\n<strong>Inequality notation<\/strong> uses the symbols [latex]&lt;[\/latex], [latex]&gt;[\/latex], [latex]\\le[\/latex], and [latex]\\ge[\/latex] to describe ranges of numbers.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x &gt; 3[\/latex] means all numbers greater than [latex]3[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]2 \\le x &lt; 5[\/latex] means all numbers greater than or equal to [latex]2[\/latex] and less than [latex]5[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><strong>Set-builder notation<\/strong> uses curly braces [latex]{}[\/latex] and a vertical bar [latex]|[\/latex] to describe sets based on their properties.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">[latex]{x \\mid x &gt; 3}[\/latex] means the set of all [latex]x[\/latex] such that [latex]x[\/latex] is greater than [latex]3[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]{x \\mid 2 \\le x &lt; 5}[\/latex] means the set of all [latex]x[\/latex] such that [latex]x[\/latex] is greater than or equal to [latex]2[\/latex] and less than [latex]5[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><strong>Interval notation<\/strong> uses parentheses [latex]([\/latex] [latex])[\/latex] and square brackets [latex][[\/latex] [latex]][\/latex] to represent continuous ranges of numbers.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">[latex](3, \\infty)[\/latex] means all numbers greater than [latex]3[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex][2, 5)[\/latex] means all numbers greater than or equal to [latex]2[\/latex] and less than [latex]5[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>The table below compares inequality notation, set-builder notation, and interval notation.\r\n<table style=\"height: 312px;\">\r\n<thead>\r\n<tr style=\"height: 30px;\">\r\n<th style=\"height: 30px; width: 194px;\"><\/th>\r\n<th style=\"height: 30px; width: 148px;\">Inequality Notation<\/th>\r\n<th style=\"height: 30px; width: 118px;\">Set-builder Notation<\/th>\r\n<th style=\"height: 30px; width: 80px;\">Interval Notation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 45px;\">\r\n<td style=\"height: 45px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/1.png\"><img class=\"size-full wp-image-12492 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193534\/1.png\" alt=\"1\" width=\"265\" height=\"60\" \/><\/a><\/td>\r\n<td style=\"height: 45px; width: 148px;\">[latex]5 \\lt h \\le 10[\/latex]<\/td>\r\n<td style=\"height: 45px; width: 118px;\">[latex]\\{h | 5 &lt; h \\le 10\\}[\/latex]<\/td>\r\n<td style=\"height: 45px; width: 80px;\">[latex](5,10][\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 48px;\">\r\n<td style=\"height: 48px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/2.png\"><img class=\"size-full wp-image-12493 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193535\/2.png\" alt=\"2\" width=\"281\" height=\"75\" \/><\/a><\/td>\r\n<td style=\"height: 48px; width: 148px;\">[latex]5 \\le\u00a0 h&lt;10[\/latex]<\/td>\r\n<td style=\"height: 48px; width: 118px;\">[latex]\\{h | 5 \\le h &lt; 10\\}[\/latex]<\/td>\r\n<td style=\"height: 48px; width: 80px;\">[latex][5,10)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 49px;\">\r\n<td style=\"height: 49px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/3.png\"><img class=\"size-full wp-image-12494 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193537\/3.png\" alt=\"3\" width=\"283\" height=\"76\" \/><\/a><\/td>\r\n<td style=\"height: 49px; width: 148px;\">[latex]5 \\lt h\\lt 10[\/latex]<\/td>\r\n<td style=\"height: 49px; width: 118px;\">[latex]\\{h | 5 &lt; h &lt; 10\\}[\/latex]<\/td>\r\n<td style=\"height: 49px; width: 80px;\">[latex](5,10)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 51px;\">\r\n<td style=\"height: 51px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/4.png\"><img class=\"size-full wp-image-12495 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193538\/4.png\" alt=\"4\" width=\"271\" height=\"76\" \/><\/a><\/td>\r\n<td style=\"height: 51px; width: 148px;\">[latex]h&lt;10[\/latex]<\/td>\r\n<td style=\"height: 51px; width: 118px;\">[latex]\\{h | h &lt; 10\\}[\/latex]<\/td>\r\n<td style=\"height: 51px; width: 80px;\">[latex](-\\infty,10)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 44px;\">\r\n<td style=\"height: 44px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/5.png\"><img class=\"size-full wp-image-12496 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193540\/5.png\" alt=\"5\" width=\"310\" height=\"66\" \/><\/a><\/td>\r\n<td style=\"height: 44px; width: 148px;\">[latex]h&gt;10[\/latex]<\/td>\r\n<td style=\"height: 44px; width: 118px;\">[latex]\\{h | h &gt; 10\\}[\/latex]<\/td>\r\n<td style=\"height: 44px; width: 80px;\">[latex](10,\\infty)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 45px;\">\r\n<td style=\"height: 45px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/6.png\"><img class=\"size-full wp-image-12497 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193542\/6.png\" alt=\"6\" width=\"359\" height=\"67\" \/><\/a><\/td>\r\n<td style=\"height: 45px; width: 148px;\">All real numbers<\/td>\r\n<td style=\"height: 45px; width: 118px;\">[latex]\\mathbf{R}[\/latex]<\/td>\r\n<td style=\"height: 45px; width: 80px;\">[latex](\u2212\\infty,\\infty)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<p class=\"font-600 text-xl font-bold\"><strong>Special Cases and Symbols<\/strong><\/p>\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\infty[\/latex] (infinity) is used in interval notation to represent unbounded intervals<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The empty set is represented as [latex]\\emptyset[\/latex] or [latex]\\{\\}[\/latex] in set-builder notation, and as [latex][\u00a0 ][\/latex] in interval notation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The union of sets is represented by the symbol [latex]\\cup[\/latex]<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox example\">Describe the intervals of values shown below using inequality notation, set-builder notation, and interval notation.<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193544\/CNX_Precalc_Figure_01_02_0042.jpg\" alt=\"Line graph of 1&lt;=x&lt;=3 and 5&lt;x.\" width=\"487\" height=\"50\" \/>To describe the values, [latex]x[\/latex], included in the intervals shown, we would say, \" [latex]x[\/latex] is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.\"\r\n<table style=\"width: 102.547%;\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 19.1257%;\"><strong>Inequality<\/strong><\/td>\r\n<td style=\"width: 82.1494%;\">[latex]1\\le x\\le 3\\text{ or }x&gt;5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 19.1257%;\"><strong>Set-builder notation<\/strong><\/td>\r\n<td style=\"width: 82.1494%;\">[latex]\\left\\{x|1\\le x\\le 3 \\text{ or }x&gt;5\\right\\}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 19.1257%;\"><strong>Interval notation<\/strong><\/td>\r\n<td style=\"width: 82.1494%;\">[latex]\\left[1,3\\right]\\cup \\left(5,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nRemember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. Using a parenthesis means the boundary is not included in the set.\r\n\r\n<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]317857[\/ohm_question]<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]317858[\/ohm_question]<\/section><section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]317859[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Use interval notation<\/span><\/li>\n<\/ul>\n<\/section>\n<h2 class=\"font-600 text-xl font-bold\">Set Notation: Inequality, Set-Builder, and Interval Notations<\/h2>\n<p>In mathematics, we often need to describe sets of numbers that satisfy certain conditions. There are several ways to represent these sets, each with its own advantages and uses in different contexts. This page introduces three common notations for describing sets of numbers: inequality notation, set-builder notation, and interval notation.<\/p>\n<p><strong>Inequality notation<\/strong> uses the symbols [latex]<[\/latex], [latex]>[\/latex], [latex]\\le[\/latex], and [latex]\\ge[\/latex] to describe ranges of numbers.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]x > 3[\/latex] means all numbers greater than [latex]3[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]2 \\le x < 5[\/latex] means all numbers greater than or equal to [latex]2[\/latex] and less than [latex]5[\/latex]<\/li>\n<\/ul>\n<\/section>\n<p><strong>Set-builder notation<\/strong> uses curly braces [latex]{}[\/latex] and a vertical bar [latex]|[\/latex] to describe sets based on their properties.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]{x \\mid x > 3}[\/latex] means the set of all [latex]x[\/latex] such that [latex]x[\/latex] is greater than [latex]3[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]{x \\mid 2 \\le x < 5}[\/latex] means the set of all [latex]x[\/latex] such that [latex]x[\/latex] is greater than or equal to [latex]2[\/latex] and less than [latex]5[\/latex]<\/li>\n<\/ul>\n<\/section>\n<p><strong>Interval notation<\/strong> uses parentheses [latex]([\/latex] [latex])[\/latex] and square brackets [latex][[\/latex] [latex]][\/latex] to represent continuous ranges of numbers.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<ul>\n<li class=\"whitespace-normal break-words\">[latex](3, \\infty)[\/latex] means all numbers greater than [latex]3[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex][2, 5)[\/latex] means all numbers greater than or equal to [latex]2[\/latex] and less than [latex]5[\/latex]<\/li>\n<\/ul>\n<\/section>\n<p>The table below compares inequality notation, set-builder notation, and interval notation.<\/p>\n<table style=\"height: 312px;\">\n<thead>\n<tr style=\"height: 30px;\">\n<th style=\"height: 30px; width: 194px;\"><\/th>\n<th style=\"height: 30px; width: 148px;\">Inequality Notation<\/th>\n<th style=\"height: 30px; width: 118px;\">Set-builder Notation<\/th>\n<th style=\"height: 30px; width: 80px;\">Interval Notation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 45px;\">\n<td style=\"height: 45px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12492 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193534\/1.png\" alt=\"1\" width=\"265\" height=\"60\" \/><\/a><\/td>\n<td style=\"height: 45px; width: 148px;\">[latex]5 \\lt h \\le 10[\/latex]<\/td>\n<td style=\"height: 45px; width: 118px;\">[latex]\\{h | 5 < h \\le 10\\}[\/latex]<\/td>\n<td style=\"height: 45px; width: 80px;\">[latex](5,10][\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 48px;\">\n<td style=\"height: 48px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12493 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193535\/2.png\" alt=\"2\" width=\"281\" height=\"75\" \/><\/a><\/td>\n<td style=\"height: 48px; width: 148px;\">[latex]5 \\le\u00a0 h<10[\/latex]<\/td>\n<td style=\"height: 48px; width: 118px;\">[latex]\\{h | 5 \\le h < 10\\}[\/latex]<\/td>\n<td style=\"height: 48px; width: 80px;\">[latex][5,10)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 49px;\">\n<td style=\"height: 49px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/3.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12494 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193537\/3.png\" alt=\"3\" width=\"283\" height=\"76\" \/><\/a><\/td>\n<td style=\"height: 49px; width: 148px;\">[latex]5 \\lt h\\lt 10[\/latex]<\/td>\n<td style=\"height: 49px; width: 118px;\">[latex]\\{h | 5 < h < 10\\}[\/latex]<\/td>\n<td style=\"height: 49px; width: 80px;\">[latex](5,10)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 51px;\">\n<td style=\"height: 51px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/4.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12495 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193538\/4.png\" alt=\"4\" width=\"271\" height=\"76\" \/><\/a><\/td>\n<td style=\"height: 51px; width: 148px;\">[latex]h<10[\/latex]<\/td>\n<td style=\"height: 51px; width: 118px;\">[latex]\\{h | h < 10\\}[\/latex]<\/td>\n<td style=\"height: 51px; width: 80px;\">[latex](-\\infty,10)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 44px;\">\n<td style=\"height: 44px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/5.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12496 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193540\/5.png\" alt=\"5\" width=\"310\" height=\"66\" \/><\/a><\/td>\n<td style=\"height: 44px; width: 148px;\">[latex]h>10[\/latex]<\/td>\n<td style=\"height: 44px; width: 118px;\">[latex]\\{h | h > 10\\}[\/latex]<\/td>\n<td style=\"height: 44px; width: 80px;\">[latex](10,\\infty)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 45px;\">\n<td style=\"height: 45px; width: 194px;\"><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/6.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12497 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193542\/6.png\" alt=\"6\" width=\"359\" height=\"67\" \/><\/a><\/td>\n<td style=\"height: 45px; width: 148px;\">All real numbers<\/td>\n<td style=\"height: 45px; width: 118px;\">[latex]\\mathbf{R}[\/latex]<\/td>\n<td style=\"height: 45px; width: 80px;\">[latex](\u2212\\infty,\\infty)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<p class=\"font-600 text-xl font-bold\"><strong>Special Cases and Symbols<\/strong><\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\infty[\/latex] (infinity) is used in interval notation to represent unbounded intervals<\/li>\n<li class=\"whitespace-normal break-words\">The empty set is represented as [latex]\\emptyset[\/latex] or [latex]\\{\\}[\/latex] in set-builder notation, and as [latex][\u00a0 ][\/latex] in interval notation<\/li>\n<li class=\"whitespace-normal break-words\">The union of sets is represented by the symbol [latex]\\cup[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">Describe the intervals of values shown below using inequality notation, set-builder notation, and interval notation.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193544\/CNX_Precalc_Figure_01_02_0042.jpg\" alt=\"Line graph of 1&lt;=x&lt;=3 and 5&lt;x.\" width=\"487\" height=\"50\" \/>To describe the values, [latex]x[\/latex], included in the intervals shown, we would say, &#8221; [latex]x[\/latex] is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.&#8221;<\/p>\n<table style=\"width: 102.547%;\" summary=\"..\">\n<tbody>\n<tr>\n<td style=\"width: 19.1257%;\"><strong>Inequality<\/strong><\/td>\n<td style=\"width: 82.1494%;\">[latex]1\\le x\\le 3\\text{ or }x>5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 19.1257%;\"><strong>Set-builder notation<\/strong><\/td>\n<td style=\"width: 82.1494%;\">[latex]\\left\\{x|1\\le x\\le 3 \\text{ or }x>5\\right\\}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 19.1257%;\"><strong>Interval notation<\/strong><\/td>\n<td style=\"width: 82.1494%;\">[latex]\\left[1,3\\right]\\cup \\left(5,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Remember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. 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