# What is the range of the function f(x) = 2/(x-1)?

May 14, 2017

$y \in \mathbb{R} , y \ne 0$

#### Explanation:

$\text{rearrange f(x) making x the subject}$

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

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

$\Rightarrow x y - y = 2$

$\Rightarrow x y = 2 + y$

$\Rightarrow x = \frac{2 + y}{y}$

The denominator cannot be zero as this would make it $\textcolor{b l u e}{\text{undefined}}$ Equating the denominator to zero and solving gives the value that y cannot be.

$\Rightarrow y = 0 \leftarrow \textcolor{red}{\text{excluded value}}$

$\Rightarrow \text{range is } y \in \mathbb{R} , y \ne 0$