# What is the domain and range of y = (x + 2) /( x + 5)?

Jun 18, 2018

The domain is $x \in \left(- \infty , - 5\right) \cup \left(- 5 , + \infty\right)$. The range is $y \in \left(- \infty , 1\right) \cup \left(1 , + \infty\right)$

#### Explanation:

The denominator must be $\ne 0$

Therefore,

$x + 5 \ne 0$

$\implies$, $x \ne - 5$

The domain is $x \in \left(- \infty , - 5\right) \cup \left(- 5 , + \infty\right)$

To find the range, proceed as follows :

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

$\implies$, $y \left(x + 5\right) = x + 2$

$\implies$, $y x + 5 y = x + 2$

$\implies$, $y x - x = 2 - 5 y$

$\implies$, $x \left(y - 1\right) = 2 - 5 y$

$\implies$, $x = \frac{2 - 5 y}{y - 1}$

The denominator must be $\ne 0$

Therefore,

$y - 1 \ne 0$

$\implies$, $y \ne 1$

The range is $y \in \left(- \infty , 1\right) \cup \left(1 , + \infty\right)$

graph{(x+2)/(x+5) [-26.77, 13.77, -10.63, 9.65]}