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

Jun 16, 2018

$x \in \mathbb{R} , x \ne - 4 , y \in \mathbb{R} . y \ne 0$

#### Explanation:

$\text{let } y = \frac{1}{x + 4}$

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

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

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

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

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

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

$x y + 4 y = 1$

$x y = 1 - 4 y$

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

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

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

$\left(- \infty , 0\right) \cup \left(0 , \infty\right)$
graph{1/(x+4) [-10, 10, -5, 5]}