What are Inverse Functions?

If functions f(x) and g(x) are inverses of each other, then f(x) = y only if g(y) = x. 

g(f(x)) = x

The figure given below describes a function and its inverse. This function is represented as f(x) and takes some input values and gives an output. The inverse of this function is denoted by f^-1(x). For example, let’s say f(x) = 2x. It doubles the number which is given as input, its inverse should make them half to get back the input. f^-1(x) = x/2. 

Let’s say we have a function f(x) = x². Now we are asked to find out the inverse of this function. This function is squaring its inputs, we know we need to take the square root for calculating the inverse. 

 f^-1(x) = √x²

 f^-1(x)  = ±x

We see that there are two answers possible, which one to choose? In such cases, the inverse is not possible. So, there are things we need to notice for the functions for which inverses are possible. Also, the function whose inverse exist is called invertible functions.

Condition for Inverse of a Function to Exist

For a function to have an inverse, the necessary and sufficient condition is

Function must be Bijective(One-One and Onto).

For example, let’s check the following graph for bijection.

This function has same values at two different values of input, this means function is not one-one. Thus, we can’t be able to find it’s inverse without restricting its domain.

Also read: What is an inverse function?

Inverse Functions are an important concept in mathematics. To comprehend inverse functions, we can picture a function as a box that takes in inputs and produces outputs. If a function consistently generates a red-colored object as output for any input object, we can identify that box as the initial function. The box that accepts both red and normal-colored objects as inputs and generates the original-colored objects as outputs, is called the inverse of the initial box.

In other words, if a function is an operation that produces an output for each input, the inverse function facilitates the identification of the specific input based on a given output. Let’s learn about inverse functions and all the different associated topics with them.