Question

Let f: R - R be given by f(x)=(2x+1)/3.Find fof and show that f is invertible?​

Answer 1

Note :

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

→ A function is said to be one-one , if

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

→ A function is said to be onto , if

Range = Co-domain

→ fof(x) = f [ f(x) ]

Solution :

Given function :

f(x) = (2x + 1)/3

★ Whether f(x) is one-one :-

Let f(x1) = f(x2)

=> (2x1 + 1)/3 = (2x2 + 1)/3

=> 2x1 + 1 = 2x2 + 1

=> 2x1 = 2x2

=> x1 = x2

=> f(x) is one-one function .

★ Whether f(x) is onto :-

Let y = f(x)

=> y = (2x + 1)/3

=> 3y = 2x + 1

=> 2x = 3y - 1

=> x = (3y - 1)/2

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

=> Range = R

=> Range = Co-domain

→ Since , f(x) is one-one onto function , thus f(x) is invertible .

Also ,

fof(x) = (4x + 5)/9

( For fof(x) , please refer to the attachment )