{"id":9822,"date":"2022-01-28T01:49:59","date_gmt":"2022-01-27T20:19:59","guid":{"rendered":"https:\/\/mathemerize.com\/?p=9822"},"modified":"2022-01-28T01:50:03","modified_gmt":"2022-01-27T20:20:03","slug":"onto-function-surjection-definition-and-examples","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/onto-function-surjection-definition-and-examples\/","title":{"rendered":"Onto Function (Surjection) -Definition and Examples"},"content":{"rendered":"

Here you will learn what is onto function with definition and examples.<\/p>\n

Let’s begin –<\/p>\n

Onto Function (Surjection) Definition<\/h2>\n

Definition<\/strong> : A function f : A \\(\\rightarrow\\) B is said to be an onto function if every element of B is the f-image of some element of A i.e. , if f(A) = B or range of f is the codomain of f.<\/p>\n

Thus, f : A \\(\\rightarrow\\) B is a surjection iff for each b \\(\\in\\) B, there exist a \\(\\in\\) A such that f(a) = b<\/p><\/blockquote>\n

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

Example<\/strong><\/span> : Let g : X \\(\\rightarrow\\) Y be the function represented by the following diagram :<\/p>\n

\"onto<\/p>\n

Solution<\/strong><\/span> : Under function g every element in Y has its pre-image X. So, g : X \\(\\rightarrow\\) Y is onto.<\/p>\n

Algorithm to Check for Onto<\/h2>\n

Let f : A \\(\\rightarrow\\) B be the given function.<\/p>\n

1).  Choose an arbitrary element y in b.<\/p>\n

2).  Put f(x) = y<\/p>\n

3).  Solve the equation f(x) = y for x and obtain x in terms of y. Let x = g(y).<\/p>\n

4).  If for all values of y \\(\\in\\) B, the values of x obtained from x = g(y) are in A, then f is onto.<\/p><\/blockquote>\n

Note<\/strong> : If range is same as codomain, then f is onto function.<\/p>\n

Example<\/strong><\/span> : Let f : R \\(\\rightarrow\\) R given by f(x) = \\(x^3 + 2\\) for all x \\(\\in\\) R. Then, find whether it is onto or not.<\/p>\n

Solution<\/strong><\/span> : Let y be the arbitrary element of R. Then,<\/p>\n

f(x) = y \\(\\implies\\) \\(x^3 + 2\\) = y \\(\\implies\\) x = \\((y – 2)^{1\/3})\\)<\/p>\n

Clearly, for all y \\(\\in\\) R, \\((y – 2)^{1\/3})\\) is a real number. Thus, for all y \\(\\in\\) R (co-domain) there exist x = \\((y – 2)^{1\/3})\\) in R (domain) such that f(x) = \\(x^3 + 2\\) = y.<\/p>\n

Hence, f : R \\(\\rightarrow\\) R is an onto function.<\/p>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is onto function with definition and examples. Let’s begin – Onto Function (Surjection) Definition Definition : A function f : A \\(\\rightarrow\\) B is said to be an onto function if every element of B is the f-image of some element of A i.e. , if f(A) = B or …<\/p>\n

Onto Function (Surjection) -Definition and Examples<\/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":[367,835,836],"yoast_head":"\nOnto Function (Surjection) -Definition and Examples<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is onto function with definition and examples based on it.\" \/>\n<meta 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