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.

Let y = x² (say f(x))

        => x = +√y

But x is positive, as domain of f is [0, α)

         =>  x = + √y

        Therefore Inverse is y = √x = g(x)

Figure (A)

        f(g(x)) = f(√x) = x, x> 0

        g(f(x)) = g(x²) = √x² = x, x > 0

i.e. if f and g are inverse of each other then f(g(x)) = g(f(x)) = x

Illustration: How are the graphs of function and the inverse function related? These graphs are mirror images of each other about the line y = x.

Solution:

Also, if the graph of y = f(x) and y = f^-1 (x), they intersect at the point where y meet the line y = x.

Figure (B)

Graphs of the function and its inverse are shown in figures given above as Figure (A) and (B)

For Figure (A)

        y = f(x) = x²; f : [0, ∞) → [0, ∞)

Examples

1.     Define y = f(x) = x² in some other ay so that its inverse is possible.

2.     What is the inverse of y = log_e (x + √(x² + 1))

Ans.1         f :  (-α, 0]  →  [0, α)

                y = f(x) = x² and its inverse is

                y = -√x     (Figure B)

Ans.2         y = (e^x - e^x)/2