Invertible Function

Let us define a function y = f(x): X → Y. If we define a function g(y) such that x = g(y) then g is said to be the inverse function of 'f'.

Think: If f is many-to-one, g : Y → X will not satisfy the definition of a function.

So to define the inverse of a function, it must be one-one.

Further if f : X → Y is into then there must be a point in Y for which  there is no x. This again violates the definition of function for 'g' (In fact when f is one tone and onto then 'g' can be defined from range of f to domain of i.e. g : f(X) → X.

Hence, the inverse of a function can be defined within the same sets   for x and Y only when it is one-one and onto or Bijective.  

Note: A monotonic function i.e. bijection function is always invertible.

Illustration: Let f : R → R be defined as

                  y = f(x) = x². Is it invertible?

Solution:

No it is not invertible because this is a many one into function

This is many-one because for x = + a, y = a², this is into because y does not take the negative real values.

Illustration: Let f : R → [0, α) be defined as y = f(x) = x². Is it invertible?
(see figure below)

Solution:

No it is not invertible, it because it is many one onto function.

Illustration: Let f : [0, α) → [0, α) be defined as y = f(x) = x². Is it invertible? If so find its inverse.

Solution:

Yes, it is invertible because this is bijection function. Its graph is shown in figure given below.