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

Jul 4, 2017

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

#### Explanation:

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

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

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

To find any excluded value in the range rearrange f(x) making x the subject.

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

$\Rightarrow y \left(x - 2\right) = x + 2 \leftarrow \textcolor{b l u e}{\text{ cross multiplying}}$

$\Rightarrow x y - 2 y = x + 2 \leftarrow \textcolor{b l u e}{\text{ distributing}}$

$\Rightarrow x y - x = 2 + 2 y \leftarrow \textcolor{b l u e}{\text{ collecting terms in x}}$

$\Rightarrow x \left(y - 1\right) = 2 + 2 y \leftarrow \textcolor{b l u e}{\text{ factoring}}$

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

$\text{the denominator cannot equal zero}$

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

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