{"id":184452,"date":"2025-01-21T05:18:26","date_gmt":"2025-01-21T05:18:26","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=184452"},"modified":"2025-01-21T05:18:28","modified_gmt":"2025-01-21T05:18:28","slug":"for-each-function-graphed-below-state-whether-it-is-one-to-one","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/21\/for-each-function-graphed-below-state-whether-it-is-one-to-one\/","title":{"rendered":"For each function graphed below, state whether it is one-to-one"},"content":{"rendered":"\n<p>For each function graphed below, state whether it is one-to-one.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/01\/image-331.png\" alt=\"\" class=\"wp-image-184453\"\/><\/figure>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To determine whether a function is <strong>one-to-one<\/strong>, we need to verify if <strong>each output corresponds to exactly one input<\/strong>. This can be done using the <strong>horizontal line test<\/strong>: if any horizontal line intersects the graph of the function at more than one point, the function is not one-to-one.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Analysis of the Graphed Functions:<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Function 1 (Graph 1)<\/strong>:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The graph in the first image shows a parabolic curve opening upwards.<\/li>\n\n\n\n<li><strong>Horizontal Line Test<\/strong>: A horizontal line intersects the graph at two points (except at the vertex).<\/li>\n\n\n\n<li><strong>Conclusion<\/strong>: This function is <strong>not one-to-one<\/strong>, as there are multiple inputs for certain outputs.<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Function 2 (Graph 2)<\/strong>:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The graph in the second image appears to be a straight line with a positive slope.<\/li>\n\n\n\n<li><strong>Horizontal Line Test<\/strong>: A horizontal line intersects the graph at only one point everywhere.<\/li>\n\n\n\n<li><strong>Conclusion<\/strong>: This function is <strong>one-to-one<\/strong>, as each output corresponds to only one input.<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Function 3 (Graph 3)<\/strong>:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The graph in the third image looks like a sine wave.<\/li>\n\n\n\n<li><strong>Horizontal Line Test<\/strong>: A horizontal line intersects the graph at multiple points periodically.<\/li>\n\n\n\n<li><strong>Conclusion<\/strong>: This function is <strong>not one-to-one<\/strong>, as some outputs have multiple inputs.<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Function 4 (Graph 4)<\/strong>:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The graph in the fourth image shows an increasing exponential curve.<\/li>\n\n\n\n<li><strong>Horizontal Line Test<\/strong>: A horizontal line intersects the graph at only one point everywhere.<\/li>\n\n\n\n<li><strong>Conclusion<\/strong>: This function is <strong>one-to-one<\/strong>, as each output corresponds to only one input.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation (300 Words):<\/h3>\n\n\n\n<p>A <strong>one-to-one function<\/strong> ensures each value in the range is uniquely paired with one value in the domain. For a visual determination, we apply the <strong>horizontal line test<\/strong>. If a horizontal line crosses the graph at more than one point, the function fails to be one-to-one.<\/p>\n\n\n\n<p>For instance, the <strong>parabola (Graph 1)<\/strong> is symmetric about its vertex, so there are pairs of inputs (x-values) with the same output (y-value), failing the test. Similarly, the <strong>sine wave (Graph 3)<\/strong> is periodic, meaning certain y-values are repeated across cycles. Hence, it is also not one-to-one.<\/p>\n\n\n\n<p>Conversely, a <strong>linear function (Graph 2)<\/strong> with a non-zero slope always passes the horizontal line test since there is a consistent one-to-one mapping of x to y. Similarly, an <strong>exponential function (Graph 4)<\/strong> is strictly increasing (or decreasing, depending on the base), ensuring unique outputs for each input.<\/p>\n\n\n\n<p>Understanding one-to-one functions is crucial in mathematics since they guarantee the existence of an inverse function, which reverses the mapping. For practical applications, one-to-one functions are vital in contexts like cryptography, where unique input-output relationships are essential.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>For each function graphed below, state whether it is one-to-one. The Correct Answer and Explanation is : To determine whether a function is one-to-one, we need to verify if each output corresponds to exactly one input. This can be done using the horizontal line test: if any horizontal line intersects the graph of the function [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-184452","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/184452","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=184452"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/184452\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=184452"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=184452"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=184452"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}