# What is the inverse function?

Sep 25, 2015

If $f$ is a function, then the inverse function, written ${f}^{- 1}$, is a function such that ${f}^{- 1} \left(f \left(x\right)\right) = x$ for all $x$.

#### Explanation:

For example, consider the function:

$f \left(x\right) = \frac{2}{3 - x}$

(which is defined for all $x \ne 3$)

If we let $y = f \left(x\right) = \frac{2}{3 - x}$, then we can express $x$ in terms of $y$ as:

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

This gives us a definition of ${f}^{-} 1$ as follows:

${f}^{- 1} \left(y\right) = 3 - \frac{2}{y}$

(which is defined for all $y \ne 0$)

Then ${f}^{- 1} \left(f \left(x\right)\right) = 3 - \frac{2}{f} \left(x\right) = 3 - \frac{2}{\frac{2}{3 - x}} = 3 - \left(3 - x\right) = x$