# How do you find the domain and range of y= 1/(x+4) -1?

Jun 16, 2018

$x \in \mathbb{R} , x \ne - 4 , y \in \mathbb{R} , y \ne - 1$

#### Explanation:

$\text{we can express y as}$

$y = \frac{1}{x + 4} - \frac{x + 4}{x + 4} = \frac{- x - 3}{x + 4}$

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

$\text{solve "x+4=0rArrx=-4larrcolor(red)"excluded value}$

$\text{domain } x \in \mathbb{R} , x \ne - 4$

$\left(- \infty , - 4\right) \cup \left(- 4 , \infty\right) \leftarrow \textcolor{b l u e}{\text{in interval notation}}$

$\text{to find range rearrange making x the subject}$

$y \left(x + 4\right) = - x - 3$

$x y + 4 y = - x - 3$

$x y + x = - 3 - 4 y$

$x \left(y + 1\right) = - \left(3 + 4 y\right)$

$x = - \frac{3 + 4 y}{y + 1}$

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

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

$\left(- \infty , - 1\right) \cup \left(- 1 , \infty\right) \leftarrow \textcolor{b l u e}{\text{in interval notation}}$
graph{1/(x+4)-1 [-10, 10, -5, 5]}