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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)

  figure-invertible-or-not

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.

     graph-for-invertible-function                        

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)
 
                                             f-and-g-are-inverse

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.

                Graphs-of-function-and-its-inverse

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 = loge (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 = (ex - ex)/2

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