Inverse of a Function

(A) Consider the relation   

This is  many-to-one function. Now let us find the Inverse of this relation.

Pictorially, it can be represented as   

Clearly this relation does not represent a function because the element of set B  does not

have unique image in  set A .

(B) Now take another relation   

It represents one-to-one onto function. Now let us find the Inverse  of this relation,

which is represented pictorially as   

This represent a function.

(C) Consider the relation  

It represents many-to-one Into function . Now find the Inverse  of the relation.
Pictorially it is represented as   

This does not represent a function, because element 6 of  set B  is not associated with any 
element of  set A  . Also note that the elements of   set B  does not have unique image.

(D) Let us take the following relation   

It represent one-to-one Into function. Find the Inverse of the relation.   

It does not represent a function because the element  7  of  set B  is not associated with any

element of  set A . 

From above relations we see that we may or may not get a relation as a function where we

find the INVERSE of a relation (function).

We see that the Inverse of a function exists only if the function is one-to-one onto 

function i.e,  only if it is a  Bijective function.