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Consider F : R → R Given by F(X) = 4x + 3. Show that F is Invertible. Find the Inverse of F. - Mathematics

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Question

Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.

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Solution

Injectivity of f :
Let x and y be two elements of domain (R), such that
f(x) = f(y)
⇒ 4x + 3 = 4y + 3
⇒ 4x = 4y
⇒ x = y
So, f is one-one.

Surjectivity of f :                 
Let y be in the co-domain (R), such that f(x) = y.

⇒ 4x + 3 = y 

⇒ 4x = y -3

⇒ `x = (y-3)/4 in ` R (domain)

⇒ f is onto.
So, f is a bijection and, hence, is invertible.

Finding f -1

 Let f-1 (x) = y                ....... (1)

⇒ x = f (y)

⇒ x = 4y + 3

⇒ x − 3 = 4y

⇒ `y = (x -3)/4`

So, `f^-1 (x) = (x-3)/4`          [ from (1) ]

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Chapter 2: Functions - Exercise 2.4 [Page 68]

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RD Sharma Mathematics [English] Class 12
Chapter 2 Functions
Exercise 2.4 | Q 6 | Page 68

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