# How do you find the domain and range of f(x)= 2/(x-5)?

Jan 18, 2018

The domain is $x \in \mathbb{R} - \left\{5\right\}$. The range is $y \in \mathbb{R} - \left\{0\right\}$

#### Explanation:

As you cannot divide by $0$, the denominator is $\ne 0$

$x - 5 \ne 0$, $\implies$, $x \ne 5$

The domain is $x \in \mathbb{R} - \left\{5\right\}$

To find the range, proceed as follows :

Let $y = \frac{2}{x - 5}$

Rearranging

$y \left(x - 5\right) = 2$

$y x - 5 y = 2$

$y x = 2 + 5 y$

$x = \frac{2 + 5 y}{y}$

Here,

$y \ne 0$

The range is $y \in \mathbb{R} - \left\{0\right\}$

graph{2/(x-5) [-18.02, 18.03, -9.01, 9.01]}

Jan 18, 2018

$x \in \mathbb{R} , x \ne 5$
$y \in \mathbb{R} , y \ne 0$

#### 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.

$\text{solve "x-5=0rArrx=5larrcolor(red)"excluded value}$

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

$\text{to obtain the range rearrange making x the subject}$

$y = \frac{2}{x - 5}$

$\Rightarrow y \left(x - 5\right) = 2$

$\Rightarrow x y - 5 y = 2$

$\Rightarrow x y = 2 + 5 y$

$\Rightarrow x = \frac{5 + 2 y}{y}$

$\Rightarrow y = 0 \leftarrow \textcolor{red}{\text{is the excluded value}}$

$\Rightarrow \text{range is } y \in \mathbb{R} , y \ne 0$