 # Inverse Function Calculator

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 $$f^{-1}$$.

### Input

The inputs of the calculator are

1. 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.

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

### Output

The outputs of the calculator is

1. The entered function
2. The inverse of the given function

## Inverse of a Function

Let f be a function whose domain is the set X, and whose codomain is the set Y. Then f is invertible if and only if there exists a function $$f^{-1}$$  with domain Y and codomain X, 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 :

$$y=\sqrt{x-3}\\x=\sqrt{y-3}$$

by squaring it on both sides

$$x^{2}=y-3\\x^{2} + 3=y\\f^{-1}(x) = x^{3}+3$$

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