**Inverse Function Calculator** is an online tool to calculate the inverse of any given function. An Inverse Function also called anti function is a function which “reverses” any real-valued function f(x) and is denoted by \(f^{-1}\).

## About Inverse Function Calculator

### Input

The inputs of the calculator are

- The function for which you want to find an inverse

### Steps

Inverse function calculator is a user-friendly tool. The following is the detailed step-by-step process to find the inverse of any function.

- Enter any function in the respective input field against the text “
*Inverse function of*“ - Click on submit button to formulate the inverse of that function.
- The output i.e the inverse function is displayed in a separate window.

### Output

The outputs of the calculator is

- The entered function
- The inverse of the given function

## Inverse of a Function

Let

fbe a function whose domain is the setX, and whose codomain is the setY. Thenfisinvertibleif and only if there exists a function \(f^{-1}\) with domainYand codomainX, with the property \( f(x)=y \Leftrightarrow f^{-1}(y)=x.\)

In other words, if **f** is any function with input **x** that gives an output **y**, then its inverse function, denoted by, **\(f^{-1}\)** is applied to **y **to give output as **x**.

However, **not all functions are invertible**. The inverse of a function exists only when every element \(y \in Y\) must correspond to no more than one element \(x \in X\).

For every **injection** i.e \(f:X \rightarrow Y\) , it is necessary for the **surjection** i.e \(f^{-1}:Y \rightarrow X\) to exist in order to compute the inverse of a function. Such functions whose both injection and surjection exists is called a **Bijective function**.

In another convention namely, the **Set-theoretic** or the **Graph** definition, the functions are defined using **ordered pairs**. In this convention, the codomain and the image of the function are made similar.

**Self Inverse Function**

For a function \(f:X \rightarrow X\), If X is a set, then its inverse function is the identity function on X \(id^{-1}_{x}=id_{x}\).

**Example:**

Find the inverse of \(f(x)=\sqrt{x-3}\)

**Solution : **

by squaring it on both sides

\( x^{2}=y-3\\x^{2} + 3=y\\f^{-1}(x) = x^{3}+3\)Checkout More Calculators Here