# How do you find the domain and range of  1/(x-7)?

Apr 12, 2017

$x \in \mathbb{R} , x \ne 7$
$y \in \mathbb{R} , y \ne 0$

#### Explanation:

$\text{let } y = \frac{1}{x - 7}$

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

$\text{solve " x-7=0rArrx=7larrcolor(red)" excluded value}$

$\Rightarrow \text{ domain is } x \in \mathbb{R} , x \ne 7$

$\text{Rearrange the function to make x the subject}$

$\Rightarrow y \left(x - 7\right) = 1$

$\Rightarrow x y - 7 y = 1$

$\Rightarrow x y = 1 + 7 y$

$\Rightarrow x = \frac{1 + 7 y}{y}$

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

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