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

Jun 9, 2017

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

#### Explanation:

$\text{the denominator of y cannot be zero as this would make y}$
$\textcolor{b l u e}{\text{undefined}}$

$\text{equating the denominator to zero and solving gives the value}$
$\text{that x cannot be}$

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

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

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

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

$\Rightarrow x y + 5 y = x$

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

$\Rightarrow x \left(y - 1\right) = - 5 y$

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

$\text{the denominator cannot be zero}$

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

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