Theorems on Inverse of a Function

In this class, We discuss Theorems on Inverse of a Function.

The reader should have prior knowledge of the inverse of a function. Click Here.

Theorem 1:

Show the inverse of a one-to-one and onto function, which is one-to-one and onto.

The below diagram shows a one-to-one and onto function.

f(x) = y

we can write x = f'(y).

Given that f is a one-to-one function.

Assume f'(y1) = f'(y2) = x

y1 = f(x)

y2 = f(x)

y1 = f(x) = y2

y1 map to x and y2 map to x, which is not possible. Because f is one-to-one.

y1 = y2

Given f is onto function.

f(x) = y

from given we can write x = f'(y)

for any x there is a mapping in f’. So the range of f’ = x

f’ is onto function.

Identity function:

A mapping from X -> X is identity mapping. if f(x) = x.

Theorem 2:

If f X -> Y is invertible, then show the below.

f’ o f = Ix

f o f’ = Iy

Solution:

Let x be an element in X

y = f(x)

x = f'(y)

f’of(x) = f'(f(x))

f'(y)

= x

f’of(x) = x = Ix

Similarly, y is an element in Y

fof'(y) = f(f'(y))

f(x)

=y

fof'(y) = y = Iy