Question

Consider f:R_R given by f(x)=5x-3.show that f is invertible and find the inverse

Answer 1

Answer :

Inverse function : g(x) = (x + 3)/5

Note :

★ A function f(x) is said to be invertible if it is one-one onto function .

★ One-One function : A function f(x) is said to be one-one if

f(x1) = f(x2) => x1 = x2 .

★ Onto function : A function f(x) is said to be onto function if Range = Co-domain .

Solution :

Given function :

f : R → R , f(x) = 5x - 3:

★ Whether f(x) is one-one :-

Let f(x1) = f(x2)

=> 5x1 - 3 = 5x2 - 3

=> 5x1 = 5x2

=> x1 = x2

Since , f(x1) = f(x2) => x1 = x2 , thus the given function f(x) is one-one .

★ Whether f(x) is onto :-

Let y = f(x)

=> y = 5x - 3

=> 5x = y + 3

=> x = (y + 3)/5

For x to be real , y can be any real number .

=> Range = R

=> Range = Co-domain

Since , Range (f) = Co-domain (f) , thus the given function f(x) is onto .

Clearly ,

The function f(x) is one-one onto , thus it is invertible .

Also ,

The inverse of the given function f(x) will be given as ;

g(x) = (x + 3)/5