# How do you find the domain and range of y = { x/ (x + 5) }?

Aug 19, 2017

$x \in \mathbb{R} , x \ne - 5$
$y \in \mathbb{R} , y \ne 1$

#### Explanation:

The denominator of y cannot be zero as this would make y undefined. Equating the denominator to zero and solving gives the value that x cannot be.

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

$\Rightarrow \text{domain is } x \in \mathbb{R} , x \ne - 5$

To find any excluded values in the range rearrange making x the subject.

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

$\Rightarrow x y + 5 y = x$

$\Rightarrow x y - x = - 5 y \leftarrow \textcolor{b l u e}{\text{ collect term in x together}}$

$\Rightarrow x \left(y - 1\right) = - 5 y \leftarrow \textcolor{b l u e}{\text{ factor out x}}$

$\Rightarrow x = - 5 \frac{y}{y - 1}$

$\text{the denominator cannot equal zero}$

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

$\Rightarrow \text{range is } y \in \mathbb{R} , y \ne 1$