# How do you find the domain and range of f(x)=(x-2)/(x+4) ?

Apr 22, 2018

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

#### Explanation:

$\text{let } y = \frac{x - 2}{x + 4}$

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

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

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

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

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

$\Rightarrow y \left(x + 4\right) = x - 2$

$\Rightarrow x y + 4 y - x + 2 = 0$

$\Rightarrow x \left(y - 1\right) = - 2 - 4 y$

$\Rightarrow x = - \frac{2 + 4 y}{y - 1}$

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

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

$y \in \left(- \infty , 1\right) \cup \left(1 , \infty\right)$
graph{(x-2)/(x+4) [-10, 10, -5, 5]}