{"id":553,"date":"2025-07-10T18:45:46","date_gmt":"2025-07-10T18:45:46","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=553"},"modified":"2026-01-27T15:35:57","modified_gmt":"2026-01-27T15:35:57","slug":"functions-and-function-notation-learn-it-2-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/functions-and-function-notation-learn-it-2-2\/","title":{"raw":"Functions and Function Notation: Learn It 2","rendered":"Functions and Function Notation: Learn It 2"},"content":{"raw":"<h2>Using the Vertical Line Test<\/h2>\r\n<p id=\"fs-id1165137737620\">The <strong>vertical line test<\/strong> can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does <em>not<\/em> define a function because a function has only one output value for each input value.<\/p>\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010535\/CNX_Precalc_Figure_01_01_0122.jpg\" alt=\"Three graphs visually showing what is and is not a function.\" width=\"975\" height=\"271\" \/>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<div>\r\n<h3>vertical line test<\/h3>\r\nDetermines whether a graph represents a function. To \"pass\" the vertical line test, a curve should only intersect a vertical line once for every input.\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Which of the graphs represent(s) a function [latex]y=f\\left(x\\right)?[\/latex]<span id=\"fs-id1165137786563\">\r\n<\/span><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005018\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" \/>[reveal-answer q=\"574837\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"574837\"]\r\n<p id=\"fs-id1165137629350\">The first two graphs are both functions. The third graph does not represent a function because, at most <em>x<\/em>-values, a vertical line would intersect the graph at more than one point.<\/p>\r\n\r\n<figure id=\"Figure_01_01_016\" class=\"small\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005018\/CNX_Precalc_Figure_01_01_016.jpg\" alt=\"Graph of a circle.\" width=\"487\" height=\"445\" \/><\/figure>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p id=\"fs-id1165135210137\">[ohm_question hide_question_numbers=1]317456[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">To determine if a relation is a function:\r\n<ul>\r\n \t<li>From points, a table, or a mapping: [latex]x[\/latex] cannot repeat for different values of [latex]y[\/latex]<\/li>\r\n \t<li>From a graph: use the vertical line test<\/li>\r\n<\/ul>\r\n<\/section>","rendered":"<h2>Using the Vertical Line Test<\/h2>\n<p id=\"fs-id1165137737620\">The <strong>vertical line test<\/strong> can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does <em>not<\/em> define a function because a function has only one output value for each input value.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010535\/CNX_Precalc_Figure_01_01_0122.jpg\" alt=\"Three graphs visually showing what is and is not a function.\" width=\"975\" height=\"271\" \/><\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<div>\n<h3>vertical line test<\/h3>\n<p>Determines whether a graph represents a function. To &#8220;pass&#8221; the vertical line test, a curve should only intersect a vertical line once for every input.<\/p>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Which of the graphs represent(s) a function [latex]y=f\\left(x\\right)?[\/latex]<span id=\"fs-id1165137786563\"><br \/>\n<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005018\/CNX_Precalc_Figure_01_01_013abc.jpg\" alt=\"Graph of a polynomial.\" width=\"975\" height=\"418\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q574837\">Show Solution<\/button><\/p>\n<div id=\"q574837\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137629350\">The first two graphs are both functions. The third graph does not represent a function because, at most <em>x<\/em>-values, a vertical line would intersect the graph at more than one point.<\/p>\n<figure id=\"Figure_01_01_016\" class=\"small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005018\/CNX_Precalc_Figure_01_01_016.jpg\" alt=\"Graph of a circle.\" width=\"487\" height=\"445\" \/><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p id=\"fs-id1165135210137\"><iframe loading=\"lazy\" id=\"ohm317456\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=317456&theme=lumen&iframe_resize_id=ohm317456&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">To determine if a relation is a function:<\/p>\n<ul>\n<li>From points, a table, or a mapping: [latex]x[\/latex] cannot repeat for different values of [latex]y[\/latex]<\/li>\n<li>From a graph: use the vertical line test<\/li>\n<\/ul>\n<\/section>\n","protected":false},"author":13,"menu_order":7,"template":"","meta":{"_candela_citation":"[]","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":[],"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\/553"}],"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\/13"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/553\/revisions"}],"predecessor-version":[{"id":5434,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/553\/revisions\/5434"}],"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\/553\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=553"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=553"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=553"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=553"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}