{"id":3980,"date":"2021-08-12T11:30:10","date_gmt":"2021-08-12T11:30:10","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3980"},"modified":"2022-01-28T01:48:24","modified_gmt":"2022-01-27T20:18:24","slug":"one-one-and-onto-function","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/one-one-and-onto-function\/","title":{"rendered":"One One and Onto Function (Bijection) – Definition and Examples"},"content":{"rendered":"

Here, you will learn one one and onto function (bijection) with definition and examples.<\/p>\n

Let’s begin –<\/p>\n

What is Bijection Function (One-One Onto Function) ?<\/h2>\n

Definition<\/strong> : A function f : A \\(\\rightarrow\\) B is a bijection if it is one-one as well as onto.<\/p>\n

In other words, a function f : A \\(\\rightarrow\\) B is a bijection, if it is<\/p>\n

\n

(i) one-one i.e. f(x) = f(y) \\(\\implies\\) x = y for all x, y \\(\\in\\) A.<\/p>\n

(ii) onto i.e. for all y \\(\\in\\) B, there exist x \\(\\in\\) A such that f(x) = y.<\/p>\n<\/blockquote>\n

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

Example<\/strong><\/span> : Let f : A \\(\\rightarrow\\) B be a function represented by the following diagram :<\/p>\n

\"bijection<\/p>\n

Solution<\/strong><\/span> : Clearly, f is a bijection since it is both one-one (injective) and onto (surjective).<\/p>\n

Example<\/strong><\/span> : Prove that the function f : Q \\(\\rightarrow\\) Q given by f(x) = 2x – 3 for all x \\(\\in\\) Q is a bijection.<\/p>\n

Solution<\/strong><\/span> : We observe the following properties of f.<\/p>\n

One-One (Injective)<\/strong> : Let x, y be two arbitrary elements in Q. Then,<\/p>\n

f(x) = f(y) \\(\\implies\\) 2x – 3 = 2y – 3 \\(\\implies\\) 2x = 2y \\(\\implies\\) x = y<\/p>\n

Thus, f(x) = f(y) \\(\\implies\\) x = y for all x, y \\(\\in\\) Q.<\/p>\n

So, f is one-one.<\/p>\n

Onto (Surjective) :\u00a0<\/strong>Let y be an arbitrary element of Q. Then,<\/p>\n

f(x) = y \\(\\implies\\) 2x – 3 = y \\(\\implies\\) x = \\(y + 3\\over 2\\)<\/p>\n

Clearly, for all y \\(\\in\\) Q, x = \\(y + 3\\over 2\\) \\(\\in\\) Q. Thus, for all y \\(\\in\\) Q (co-domain) there exist x \\(\\in\\) Q (domain) given by x = \\(y + 3\\over 2\\) such that f(x) = f(\\(y + 3\\over 2\\)) = 2(\\(y + 3\\over 2\\)) – 3 = y. That is every element in the co-domain has its pre-image in x.<\/p>\n

So, f is onto function.<\/p>\n

Hence, f : Q \\(\\rightarrow\\) Q is a bijection.<\/p>\n

Example<\/strong><\/span> : Let f : R \\(\\rightarrow\\) R be a function defined as f(x) = \\(2x^3 + 6x^2\\) + 12x +3cosx – 4sinx; then f is-<\/p>\n

Solution<\/strong><\/span> : We have f(x) = \\(2x^3 – 6x^2\\) + 12x + 3cosx – 4sinx<\/p>\n

\\(\\implies\\) f'(x) = \\(6x^2 – 12x\\) + 12 – 3sinx – 4cosx<\/p>\n

\\(\\implies\\) \\(6x^2 – 12x\\) + 12 = 6(\\(x-1)^2\\) + 6 = g(x) and 3sinx + 4cosx = h(x)<\/p>\n

range of g(x) = [6, \\(\\infty\\))<\/p>\n

range of h(x) = [-5, 5]<\/p>\n

hence f'(x) always lies in the interval [1, \\(\\infty\\))<\/p>\n

\\(\\implies\\) f'(x) > 0<\/p>\n

Hence f(x) is increasing i.e. one-one<\/p>\n

Now x \\(\\rightarrow\\) \\(\\infty\\) \\(\\implies\\) f \\(\\rightarrow\\) \\(\\infty\\) & x \\(\\rightarrow\\) -\\(\\infty\\) \\(\\implies\\) f \\(\\rightarrow\\) -\\(\\infty\\) & f(x) is continous.<\/p>\n

hence its range is R \\(\\implies\\) f is onto so f is bijective.<\/p>\n

Note<\/strong> :<\/p>\n

(i)\u00a0 If a line parallel to x-axis cuts the graph of the functions atmost at one point, then the f is one-one.<\/p>\n

(ii)\u00a0 If any line parallel to x-axis cuts the graph of the functions atleast at two points, then f is many-one.<\/p>\n

(iii)\u00a0 If continous functions f(x) is always increasing or decreasing in whole domain, then f(x) is one-one.<\/p>\n

(iv)\u00a0 All linear functions are one-one.<\/p>\n

(v)\u00a0 All trigonometric functions in their domain are many one.<\/p>\n

(vi)\u00a0 All even degree polynomials are many one.<\/p>\n

(vii)\u00a0 Linear by linear is one-one.<\/p>\n

(viii)\u00a0 Quadratic by quadratic with no common factor is many one.<\/p>\n

(ix)\u00a0 \u00a0A polynomial function of degree even define from R \\(\\rightarrow\\) R will always be into.<\/p>\n

(x)\u00a0 A polynomial function of degree odd defined from R \\(\\rightarrow\\) R will always be onto<\/p>\n

(xi)\u00a0 Quadratic by quadratic without any common factor define from R \\(\\rightarrow\\) R is always an into function.<\/p>\n

Thus a function can be of these four types :<\/p>\n

(i)\u00a0 one-one onto (Injective and Surjective)(Also known as Bijective mapping)<\/p>\n

(ii)\u00a0 one-one into (Injective but not surjective)<\/p>\n

(iii)\u00a0 many-one onto (surjective but not injective)<\/p>\n

(iv)\u00a0 many-one into (neither surjective nor injective)<\/p>\n

\u00a0<\/p>\n\n\n

\n
Next – What is a Periodic Function<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Domain and Range of Modulus Function<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here, you will learn one one and onto function (bijection) with definition and examples. Let’s begin – What is Bijection Function (One-One Onto Function) ? Definition : A function f : A \\(\\rightarrow\\) B is a bijection if it is one-one as well as onto. In other words, a function f : A \\(\\rightarrow\\) B …<\/p>\n

One One and Onto Function (Bijection) – Definition and Examples<\/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":[840,359,366,367,839],"yoast_head":"\nOne One and Onto Function (Bijection) - Definition and Examples<\/title>\n<meta name=\"description\" content=\"In this post, you will learn one one and onto function (bijection function) with definition and examples based on it.\" \/>\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\/one-one-and-onto-function\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"One One and Onto Function (Bijection) - 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