{"id":9715,"date":"2022-01-25T23:59:57","date_gmt":"2022-01-25T18:29:57","guid":{"rendered":"https:\/\/mathemerize.com\/?p=9715"},"modified":"2022-01-25T23:59:58","modified_gmt":"2022-01-25T18:29:58","slug":"smallest-integer-function","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/smallest-integer-function\/","title":{"rendered":"Smallest Integer Function (Ceiling Function) – Graph, Domain and Range"},"content":{"rendered":"

Here you will learn what is smallest integer function definition, graph and its domain and range with examples.<\/p>\n

Let’s begin –<\/p>\n

Smallest Integer Function (Ceiling Function)<\/h2>\n

Definition<\/strong> : For any real number x, we use the symbol \\(\\lceil x \\rceil\\) to denote the smallest integer greater than or equal to x.<\/p>\n

For example<\/strong><\/span>, \\(\\lceil 4.7 \\rceil\\) = 5, \\(\\lceil -7.2 \\rceil\\) = -7, \\(\\lceil 5 \\rceil\\) = 5, \\(\\lceil 0.75 \\rceil\\) = 1 etc.<\/p>\n

The function f : R \\(\\rightarrow\\) R defined by f(x) = \\(\\lceil x \\rceil\\) for all x \\(\\in\\) R is callled the smallest integer function or the ceiling function.<\/p><\/blockquote>\n

It is also a step function.<\/p>\n

Also Read<\/strong> : Types of Functions in Maths \u2013 Domain and Range<\/a><\/p>\n

Graph of Ceiling Function<\/h2>\n

The graph of the smallest integer function is given below.<\/p>\n

\"smallest<\/p>\n

Domain and Range of Smallest Integer Function<\/h2>\n

We observe that the domain of the smallest integer function is the set R of all real numbers and its range is the set Z  of all integers.<\/p>\n

Domain<\/strong> : R<\/p>\n

Range<\/strong> : Z<\/p><\/blockquote>\n

Properties :<\/h2>\n

Following are some properties of smallest integer function.<\/p>\n

(i) \\(\\lceil -n \\rceil\\) = – \\(\\lceil n \\rceil\\) where n \\(\\in\\) Z.<\/p>\n

(ii) \\(\\lceil -x \\rceil\\) = – \\(\\lceil x \\rceil\\) + 1, where x \\(\\in\\) R – Z<\/p>\n

(iii) \\(\\lceil x + n \\rceil\\) = \\(\\lceil x \\rceil\\) + n, where x \\(\\in\\) R – Z and n \\(\\in\\) Z<\/p>\n

(iv) \\(\\lceil x \\rceil\\) + \\(\\lceil -x \\rceil\\) = {1, if x \\(\\notin\\) Z and 0, if x \\(\\in\\) Z}<\/p>\n

(v) \\(\\lceil x \\rceil\\) + \\(\\lceil -x \\rceil\\) = {2\\(\\lceil x \\rceil\\) – 1,  if x \\(\\notin\\) Z  and  2\\(\\lceil x \\rceil\\), if x \\(\\in\\) Z}.<\/p>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is smallest integer function definition, graph and its domain and range with examples. Let’s begin – Smallest Integer Function (Ceiling Function) Definition : For any real number x, we use the symbol \\(\\lceil x \\rceil\\) to denote the smallest integer greater than or equal to x. For example, \\(\\lceil 4.7 …<\/p>\n

Smallest Integer Function (Ceiling Function) – Graph, Domain and Range<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[26],"tags":[801,799,800,798],"yoast_head":"\nSmallest Integer Function (Ceiling Function) - Graph, Domain and Range<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is smallest integer function with definition and graph, domain and range with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/smallest-integer-function\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Smallest Integer Function (Ceiling Function) - 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