{"id":3665,"date":"2021-08-05T23:46:17","date_gmt":"2021-08-05T23:46:17","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3665"},"modified":"2021-11-24T16:13:53","modified_gmt":"2021-11-24T10:43:53","slug":"domain-and-range-of-greatest-integer-function","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/domain-and-range-of-greatest-integer-function\/","title":{"rendered":"Domain and Range of Greatest Integer Function"},"content":{"rendered":"

Here, you will learn domain and range of greatest integer function and properties of greatest integer function with example.<\/p>\n

Let’s begin –<\/p>\n

Greatest Integer Function or Floor Function<\/h2>\n

For any real number x, we use the symbol [x] or \\(\\lfloor x \\rfloor\\) to denote the greatest integer less than or equal to x. For example,<\/p>\n

[2.75] = 2, [3] =3, [0.74] = 0, [-7.45] = -8 etc.<\/p>\n

\n

The Function f : R \\(\\rightarrow\\) R defined by f(x) = [x] for all x \\(\\in\\) R<\/strong> is called the greatest integer function or the floor function.<\/p>\n<\/blockquote>\n

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

Domain and Range<\/h2>\n

Clearly. domain<\/strong> of the greatest integer function is the set of all real numbers and the range<\/strong> is the set Z of all integers as it attains only integer values.<\/p>\n

\n

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

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

The graph of the greatest integer function is shown in figure.<\/p>\n

\"greatest<\/p>\n

Greatest integer function<\/strong><\/p>\n

Properties of Greatest Integer Function<\/h2>\n

If n is an integer and x is a real number between n and n + 1, then<\/p>\n

(i) [-n] = -[n]<\/p>\n

(ii) [x + k] = [x] + k, for any integer k.<\/p>\n

(iii) [-x] = -[x] – 1<\/p>\n

(iv) [x] + [-x] = \\(\\begin{cases} -1, & \\text{if}\\ x \\notin Z \\\\
0, & \\text{if}\\ x \\in Z \\end{cases}\\)<\/p>\n

(v) [x] – [-x] = \\(\\begin{cases} 2[x] + 1, & \\text{if}\\ x \\notin Z \\\\
2[x], & \\text{if}\\ x \\in Z \\end{cases}\\)<\/p>\n

(vi) [x] \\(\\ge\\) k \\(\\implies\\) x > k, where k \\(\\in\\) Z<\/p>\n

(vii) [x] > k \\(\\implies\\) x \\(\\ge\\) k + 1, where k \\(\\in\\) Z<\/p>\n

(viii) [x] > k \\(\\implies\\) x \\(\\ge\\) k + 1, where k \\(\\in\\) Z<\/p>\n

(ix) [x] < k \\(\\implies\\) x < k, where k \\(\\in\\) Z<\/p>\n

(x) [x + y] = [x] + [y + x – [x]] for all x, y \\(\\in\\) R<\/p>\n\n\n

Example : <\/span> If y = 2[x] + 3 & y = 3[x – 2] + 5, then find [x + y] where [.] denotes greatest integer function.<\/p>\n

Solution : <\/span>y = 3[x – 2] + 5 = 3[x] – 1

\n so 3[x] – 1 = 2[x] + 3

\n [x] = 4 \\(\\implies\\) 4 \\(\\le\\) x < 5

\nthen y = 11

\nso x + y will lie in the interval [15, 16)

\nso [x + y] = 15

\n <\/p>\n\n\n

Hope you learnt domain and range of greatest integer function, learn more concepts of function and practice more questions to get ahead in the competition. Good luck!<\/p>\n\n\n

\n
Next – Domain and Range of Modulus Function<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Types of Functions in Maths<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here, you will learn domain and range of greatest integer function and properties of greatest integer function with example. Let’s begin – Greatest Integer Function or Floor Function For any real number x, we use the symbol [x] or \\(\\lfloor x \\rfloor\\) to denote the greatest integer less than or equal to x. For example, …<\/p>\n

Domain and Range of Greatest Integer Function<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","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":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[26],"tags":[370,372,373,359,374,381,375,371],"yoast_head":"\nDomain and Range of Greatest Integer Function - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post, you will learn domain and range of greatest integer function and properties of greatest integer function with example.\" \/>\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\/domain-and-range-of-greatest-integer-function\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Domain and Range of Greatest Integer Function - 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