# What is the inverse function for f(x) = x/(x-2)?

Sep 22, 2015

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

#### Explanation:

Setting $y = f \left(x\right)$

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

this may be rearranged (intermediate steps shown) as follows

$y \left(x - 2\right) = x$

$x \cdot y - 2 y = x$

$x \cdot y - x = 2 y$

$x \left(y - 1\right) = 2 y$

$x = \frac{2 y}{y - 1}$

This expression shows $x$ in terms of $y$.

That is, it is the inverse of function $f \left(x\right)$.

That is,

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

as required.