{"id":3984,"date":"2021-08-12T12:05:53","date_gmt":"2021-08-12T12:05:53","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3984"},"modified":"2022-01-16T16:54:37","modified_gmt":"2022-01-16T11:24:37","slug":"what-is-inverse-of-a-function","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/what-is-inverse-of-a-function\/","title":{"rendered":"What is Inverse of a Function – Properties and Example"},"content":{"rendered":"

Here, you will learn what is inverse of a function, its properties and how to find the inverse of a function.<\/p>\n

Let’s begin –<\/p>\n

Inverse of a Function<\/h2>\n

Let f : A \\(\\rightarrow\\) B be a one-one & onto function, then there exists a unique function g : B \\(\\rightarrow\\) A such that f(x) = y \\(\\iff\\) g(y) = x, \\(\\forall\\) x \\(\\in\\) A & y \\(\\in\\) B. Then g is said to be inverse of f.<\/p>\n

Properties of inverse functions<\/strong><\/h4>\n
\n

(a)  The inverse of bijection is unique.<\/p>\n

(b)  The inverse of bijection is also a bijection.<\/p>\n

(c)  If f & g are two bijections f : A \\(\\rightarrow\\) B, g : B \\(\\rightarrow\\) C then the inverse of gof exists and \\((gof)^{-1}\\) = \\(f^{-1}\\)o\\(g^{-1}\\).<\/p>\n<\/blockquote>\n

How to Find Inverse of Function<\/strong><\/h4>\n

In order to find the inverse of a function, we may use the following algorithm.<\/p>\n

Let f : A \\(\\rightarrow\\) B be a bijection. To find the inverse of f we follow the following steps:<\/p>\n

Step 1 : Put f(x) = y, where y \\(\\in\\) B and  x \\(\\in\\) A.<\/p>\n

Step 2 : Solve f(x) = y to obtain x in terms of y.<\/p>\n

Step 3 : In the relation obtained in step 2 replace x by \\(f^{-1}(y)\\) to obtain the required inverse of f. <\/p>\n\n\n

Example : <\/span> Let f : R \\(\\rightarrow\\) R be defined by f(x) = \\((e^x – e^{-x})\\)\/2. Is f(x) invertible?. If so, find its inverse. <\/p>\n

Solution : <\/span>Let us check the invertibility of f(x) :

\n (a) One-One :

\n f(x) = \\(1\\over 2\\)\\((e^x – e^{-x})\\) \\(\\implies\\) f'(x) = \\(1\\over 2\\)\\((e^x + e^{-x})\\)

\n f'(x) > 0, f(x) is increasing function
\n \\(\\therefore\\)   f(x) is one-one function.

\n (b) Onto :

\n As x tends to larger and larger values so does f(x) and when x \\(\\rightarrow\\) \\(\\infty\\), f(x) \\(\\rightarrow\\) \\(\\infty\\)
\n Similarly as x \\(\\rightarrow\\) -\\(\\infty\\), f(x) \\(\\rightarrow\\) -\\(\\infty\\) i.e. \n -\\(\\infty\\) < f(x) < \\(\\infty\\) so long as x \\(\\in\\) (-\\(\\infty\\), \\(\\infty\\))
\n Hence the range of f is same as the set R. Therefore f(x) is onto.

\n Since f(x) is both one-one and onto, f(x) is invertible.

\n (c) To find \\(f^{-1}\\)(x) : Interchange x & y

\n \\(1\\over 2\\)\\((e^y + e^{-y})\\) = x \\(\\implies\\) \\(e^{2y} – 2xe^{y}\\) – 1 = 0

\n \\(\\implies\\)   \\(e^y\\) = \\(2x \\pm \\sqrt{4x^2 + 4}\\over 2\\) \\(\\implies\\) \\(e^y\\) = \\(x \\pm \\sqrt{1 + x^2}\\)

\n Since \\(e^y\\) > 0, hence negative sign is ruled out and Hence \\(e^y\\) = \\(x + \\sqrt{1 + x^2}\\)

\n Taking logarithm, we have y = ln(x + \\(\\sqrt{1 + x^2}\\)) or \\(f^{-1}\\)(x) = ln(x + \\(\\sqrt{1 + x^2}\\))

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

\n
Next – Algebraic Operations on Functions<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – What is a Periodic Function<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here, you will learn what is inverse of a function, its properties and how to find the inverse of a function. Let’s begin – Inverse of a Function Let f : A \\(\\rightarrow\\) B be a one-one & onto function, then there exists a unique function g : B \\(\\rightarrow\\) A such that f(x) = …<\/p>\n

What is Inverse of a Function – Properties and Example<\/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":[362,359,361,360],"yoast_head":"\nWhat is Inverse of a Function - Properties and Example - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post, you will learn what is inverse of a function, its properties and how to find the inverse of a function.\" \/>\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\/what-is-inverse-of-a-function\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What is Inverse of a Function - 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