{"id":183951,"date":"2025-01-17T12:14:35","date_gmt":"2025-01-17T12:14:35","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=183951"},"modified":"2025-01-17T12:14:37","modified_gmt":"2025-01-17T12:14:37","slug":"use-interval-notation-to-write-the-intervals-over-which-f-is-a-increasing-b-dec-and-c-constant","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/17\/use-interval-notation-to-write-the-intervals-over-which-f-is-a-increasing-b-dec-and-c-constant\/","title":{"rendered":"Use interval notation to write the intervals over which f is (a) increasing, (b) dec and (c) constant"},"content":{"rendered":"\n<p>Use interval notation to write the intervals over which f is (a) increasing, (b) dec and (c) constant.<\/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-311.png\" alt=\"\" class=\"wp-image-183952\"\/><\/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 the intervals over which a function ( f(x) ) is increasing, decreasing, or constant, we analyze its behavior as ( x ) progresses from left to right on its graph. Here&#8217;s how to approach this:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Increasing Intervals<\/strong>: A function is increasing on an interval if, for any two points ( x_1 ) and ( x_2 ) within that interval where ( x_1 &lt; x_2 ), the corresponding function values satisfy ( f(x_1) &lt; f(x_2) ). Graphically, this means the curve ascends as it moves from left to right. In interval notation, if ( f(x) ) increases between ( x = a ) and ( x = b ), we denote this as ( (a, b) ).<\/li>\n\n\n\n<li><strong>Decreasing Intervals<\/strong>: A function is decreasing on an interval if, for any ( x_1 &lt; x_2 ) within that interval, ( f(x_1) > f(x_2) ). This indicates the graph descends as ( x ) increases. For instance, if ( f(x) ) decreases from ( x = c ) to ( x = d ), we write ( (c, d) ).<\/li>\n\n\n\n<li><strong>Constant Intervals<\/strong>: A function is constant on an interval if ( f(x_1) = f(x_2) ) for all ( x_1 ) and ( x_2 ) within that interval. This results in a horizontal line segment on the graph, indicating no change in ( f(x) ) as ( x ) varies. If ( f(x) ) remains constant between ( x = e ) and ( x = f ), we express this as ( (e, f) ).<\/li>\n<\/ol>\n\n\n\n<p><strong>Steps to Identify These Intervals from a Graph<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Observe the Graph<\/strong>: Starting from the leftmost point, trace the graph to the right, noting where it ascends, descends, or remains flat.<\/li>\n\n\n\n<li><strong>Identify Critical Points<\/strong>: Points where the graph changes direction (peaks and troughs) or flattens are crucial. These are often local maxima, minima, or points of inflection.<\/li>\n\n\n\n<li><strong>Determine Intervals<\/strong>: Based on these critical points, segment the ( x )-axis into intervals where the function is consistently increasing, decreasing, or constant.<\/li>\n<\/ul>\n\n\n\n<p><strong>Example<\/strong>:<\/p>\n\n\n\n<p>Consider a function ( f(x) ) with the following behavior:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>It increases from ( x = -5 ) to ( x = -2 ).<\/li>\n\n\n\n<li>Remains constant from ( x = -2 ) to ( x = 1 ).<\/li>\n\n\n\n<li>Increases again from ( x = 1 ) to ( x = 3 ).<\/li>\n\n\n\n<li>Decreases from ( x = 3 ) to ( x = 5 ).<\/li>\n<\/ul>\n\n\n\n<p>In interval notation:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Increasing<\/strong>: ( (-5, -2) \\cup (1, 3) )<\/li>\n\n\n\n<li><strong>Constant<\/strong>: ( (-2, 1) )<\/li>\n\n\n\n<li><strong>Decreasing<\/strong>: ( (3, 5) )<\/li>\n<\/ul>\n\n\n\n<p>It&#8217;s essential to use open intervals (parentheses) because at the exact points where the function changes direction, it is neither increasing nor decreasing. These points are transitions and don&#8217;t belong to any interval of monotonicity.<\/p>\n\n\n\n<p>By carefully analyzing the graph and noting these intervals, we can accurately describe where the function increases, decreases, or remains constant.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Use interval notation to write the intervals over which f is (a) increasing, (b) dec and (c) constant. The Correct Answer and Explanation is : To determine the intervals over which a function ( f(x) ) is increasing, decreasing, or constant, we analyze its behavior as ( x ) progresses from left to right on [&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-183951","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183951","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=183951"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183951\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=183951"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=183951"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=183951"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}