Inverse Function

Inverse Function

The reverse of a given bijective function is known as inverse function. Keeping (-1) as the power of the given function denotes the inverse of that function. Eg: $f^{-1}$

A bijective function is a one to one (injective) onto (surjective) function.

In other words, the function that we obtain after interchanging the domain and range, is said to be inverse function. 

Let us have a function: f= {(1,2),(2,3),(3,4)}
Domain of the above function = {1,2,3}
Range of the above function = {2,3,4}

Now, when we want the inverse function of ‘f’ i.e. $f^{-1}$ we interchange the position of domain and range in the function.

So, we obtain, $f^{-1}$ = {(2,1),(3,2),(4,3)}
Domain of the inverse function = {2,3,4}
Range of the inverse function = {1,2,3}

So, when we obtain an inverse function, the domain of the previous function becomes the range and the range of the previous function becomes the domain of the inverse function.

Remember: Not every inverse function of a function is necessarily a function. 

Solved Examples

Example 1: Find the inverse of the given function f={(1,1),(2,4),(3,9),(4,16),(5,25)}.

Solution:
Given,

f={(1,1),(2,4),(3,9),(4,16),(5,25)}

Now,
Interchange the positions of domain and range

$f^{-1}$={(1,1),(4,2),(9,3),(16,4),(25,5)}

Example 2: Find the inverse of the given function f(x) = 2x + 3.

Solution:
Given,

f(x) = 2x + 3
Let y = f(x) = 2x + 3
or, y = 2x + 3

Now,
Interchange the positions of domain (x) and range (y)

or, x = 2y + 3

Try to make y alone in left or right side

or, x – 3 = 2y + 3 – 3

or, x – 3 = 2y

or, $\frac{x -3}{2} = $\frac{2y}{2}$

or, $\frac{x -3}{2}$ = y

The y that you have recieved in the right side is the inverse function of f(x).

$\therefore f^{-1}(x) = \frac{x -3}{2}$