# What is the domain and range of y = (x-3)/(x+11)?

Jan 29, 2018

$x \in \mathbb{R} , x \ne - 11$
$y \in \mathbb{R} , y \ne 1$

#### Explanation:

The denominator of y cannot be zero as this would make y undefined. Equating the denominator to zero ans solving gives the value that x cannot be.

$\text{solve "x+11=0rArrx=-11larrcolor(red)"excluded value}$

$\Rightarrow \text{domain is } x \in \mathbb{R} , x \ne - 11$

$\left(- \infty , - 11\right) \cup \left(- 11 , + \infty\right) \leftarrow \textcolor{b l u e}{\text{in interval notation}}$

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

$y = \frac{\frac{x}{x} - \frac{3}{x}}{\frac{x}{x} + \frac{11}{x}} = \frac{1 - \frac{3}{x}}{1 + \frac{11}{x}}$

$\text{as } x \to \pm \infty , y \to \frac{1 - 0}{1 + 0}$

$\Rightarrow y = 1 \leftarrow \textcolor{red}{\text{excluded value}}$

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

$\left(- \infty , 1\right) \cup \left(1 , + \infty\right) \leftarrow \textcolor{b l u e}{\text{in interval notation}}$
graph{(x-3)/(x+11) [-20, 20, -10, 10]}