# Inverse Function

A function, **f**, is the inverse of another ,** f^{−1}**, if:

**f(a) = b, and f^{−1}(b) = a.**

Note that:

The domain *f ^{−1}* is the range of

*f*.

The range of *f ^{−1}* is the domain of f.

To find the range of a function we have to find the domain of its inverse function.

If two functions are the inverse of each other, their composition is the identity function.

(*f* o *f ^{ −1}*) (x) = (

*f*o

^{ −1}*f*) (

*x*) = x

The graphs of f and f^{ −1} are symmetrical about the bisector of the first and third quadrant.

**We must distinguish between the inverse function**, *f ^{−1}(x)*,

**and the inverse of a function**, .

### Calculation of the Inverse Function

1.Write the equation of the function with x and y.

2.Work out the value of the variable x as a function of the variable y.

3.The variables are exchanged.

#### Examples

Calculate the inverse function of:

Check the result for x = 2.