# How do you find the domain and range of g(x)= 2/ (x-1)?

May 1, 2017

$\text{domain } x \in \mathbb{R} , x \ne 1$

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

#### Explanation:

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

$\text{solve " x-1=0rArrx=1larrcolor(red)" excluded value}$

$\Rightarrow \text{domain } x \in \mathbb{R} , x \ne 1$

$\text{To find the excluded value/s in the range}$

$\text{Rearrange g(x) and make x the subject}$

$g \left(x\right) = y = \frac{2}{x - 1} \leftarrow \textcolor{b l u e}{\text{ cross-multiply}}$

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

$x y - y = 2$

$x y = 2 + y$

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

$\text{ the denominator cannot be zero}$

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

$\Rightarrow \text{range } y \in \mathbb{R} , y \ne 0$
graph{2/(x-1) [-10, 10, -5, 5]}