{"id":259494,"date":"2025-07-19T02:38:17","date_gmt":"2025-07-19T02:38:17","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=259494"},"modified":"2025-07-19T02:38:20","modified_gmt":"2025-07-19T02:38:20","slug":"graph-the-following-piecewise-function-and-then-find-the-domain","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/19\/graph-the-following-piecewise-function-and-then-find-the-domain\/","title":{"rendered":"Graph the following piecewise function and then find the domain"},"content":{"rendered":"\n
Graph the following piecewise function and then find the domain<\/pre>\n\n\n\n
\"\"<\/figure>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n
The correct answer is (-4,9)<\/strong>.<\/p>\n\n\n\n
Explanation:<\/h3>\n\n\n\n
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For a piecewise function, the domain is the union of the domains of all its individual pieces. The given function is:<\/p>\n\n\n\n
f(x) = { 3x\u00b2 + 1 if -4 < x < 6
{ 6 if 6 \u2264 x < 9<\/p>\n\n\n\n
We need to analyze the conditions for each piece to determine the total set of x-values.<\/p>\n\n\n\n
1. Analyze the First Piece:<\/strong><\/p>\n\n\n\n
The first rule, f(x) = 3x\u00b2 + 1, is defined for the interval specified by the inequality -4 < x < 6. This means the function follows this quadratic rule for all x-values that are strictly greater than -4 and strictly less than 6. The endpoints, -4 and 6, are not included in this portion of the domain. In interval notation, this domain is represented as (-4, 6). The parentheses signify that the endpoints are excluded.<\/p>\n\n\n\n
2. Analyze the Second Piece:<\/strong><\/p>\n\n\n\n
The second rule, f(x) = 6, is a constant function defined for the interval 6 \u2264 x < 9. This means the function has a value of 6 for all x-values that are greater than or equal to 6 but strictly less than 9. In this case, the endpoint 6 is included, which is indicated by the “less than or equal to” sign (\u2264). The endpoint 9 is excluded. In interval notation, this domain is represented as [6, 9). The square bracket [ indicates that 6 is included, while the parenthesis ) indicates that 9 is excluded.<\/p>\n\n\n\n
3. Combine the Domains:<\/strong><\/p>\n\n\n\n
To find the total domain of the entire piecewise function, we must combine, or take the union of, the two individual domains:<\/p>\n\n\n\n
Domain = (-4, 6) U [6, 9)<\/p>\n\n\n\n
The first interval, (-4, 6), covers all numbers from -4 up to 6, but not including 6. The second interval, [6, 9), starts exactly at 6 (including it) and goes up to 9 (not including it). When we combine these two sets, the gap at x = 6 from the first interval is perfectly filled by the second interval. This creates a single, continuous interval. The combined domain starts where the first interval begins and ends where the second interval stops.<\/p>\n\n\n\n
Thus, the overall domain starts at -4 (not included) and extends to 9 (not included). In interval notation, the final domain is (-4, 9)<\/strong>.<\/p>\n\n\n\n
\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"
Graph the following piecewise function and then find the domain The Correct Answer and Explanation is: The correct answer is (-4,9). Explanation: The domain of a function is the set of all possible input values (x-values) for which the function is defined. For a piecewise function, the domain is the union of the domains of all […]<\/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-259494","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/259494","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=259494"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/259494\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=259494"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=259494"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=259494"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}