## 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 asy = f(x) = x

^{2}. 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 = a

^{2}, this is into because y does not take the negative real values.

Illustration:Let f : R → [0, α) be defined as y = f(x) = x^{2}. 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) = x^{2}. 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 = x

^{2}(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(x

^{2}) = √x^{2}= x, x > 0i.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) = x

^{2}; f : [0, ∞) → [0, ∞)

Examples:1. Define y = f(x) = x

^{2}in some other ay so that its inverse is possible.2. What is the inverse of y = log

_{e}(x + √(x^{2}+ 1))Ans.1 f : (-α, 0] → [0, α)

y = f(x) = x

^{2}and its inverse isy = -√x (Figure B)

Ans.2 y = (e

^{x}- e^{x})/2

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