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

Apr 24, 2018

$x \in \mathbb{R} , x \ne - 2 , y \in \mathbb{R} , y \ne 1$

#### Explanation:

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

$\text{solve "x+2=0rArrx=-2larrcolor(red)"excluded value}$

$\Rightarrow \text{domain } x \in \mathbb{R} , x \ne - 2$

$x \in \left(- \infty , - 2\right) \cup \left(- 2 , \infty\right) \leftarrow \textcolor{b l u e}{\text{in interval notation}}$

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

$\text{For range rearrange making x the subject}$

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

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

$\Rightarrow x y - x = - 2 - 2 y$

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

$\Rightarrow x = - \frac{2 \left(1 + y\right)}{y - 1}$

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

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

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