# How do you find the domain and range of s(y) = (3y)/( y+5)?

Jun 25, 2017

$\text{domain } \in \mathbb{R} , \ne - 5$
$\text{range } \in \mathbb{R} , \ne 3$

#### Explanation:

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

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

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

$\text{to find any excluded value in the range, we can find the}$
$\text{horizontal asymptote}$

$\text{divide terms on numerator/denominator by y}$

$\frac{\frac{3 y}{y}}{\frac{y}{y} + \frac{5}{y}} = \frac{3}{1 + \frac{5}{y}}$

as $y \to \pm \infty , s \left(y\right) \to \frac{3}{1 + 0}$

$\Rightarrow y = 3 \leftarrow \textcolor{red}{\text{ is an excluded value}}$

$\text{range } \in \mathbb{R} , y \ne 3$