# How do you find the domain and range for y=2/(5(x-2))?

Aug 28, 2017

$x \in \mathbb{R} , x \ne 2$
$y \in \mathbb{R} , y \ne 0$

#### Explanation:

$\text{The denominator cannot equal zero as this would make y }$
$\text{undefined. Equating the denominator to zero and solving}$
$\text{gives the value that x cannot be}$

$\text{solve "5(x-2)=0rArrx=2larrcolor(red)" excluded value}$

$\Rightarrow \text{domain is } x \in \mathbb{R} , x \ne 2$

$\text{to find any excluded values in the range rearrange making}$
$\text{x the subject}$

$\Rightarrow y \left(5 x - 10\right) = 2 \leftarrow \textcolor{b l u e}{\text{ cross-multipling}}$

$\Rightarrow 5 x y - 10 y = 2$

$\Rightarrow 5 x y = 2 + 10 y$

$\Rightarrow x = \frac{2 + 10 y}{5 y}$

$\text{the denominator cannot equal zero}$

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

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