# What is the domain and range of y= 1/(x-10)?

Jul 17, 2017

$x \in \mathbb{R} , x \ne 10$
$y \in \mathbb{R} , y \ne 0$

#### Explanation:

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

$\text{solve "x-10=0rArrx=10larrcolor(red)" excluded value}$

$\Rightarrow \text{domain is } x \in \mathbb{R} , x \ne 10$

To find any excluded value in the range, rearrange the function making x the subject.

$\Rightarrow y \left(x - 10\right) = 1 \leftarrow \text{ cross-multiplying}$

$\Rightarrow x y - 10 y = 1 \leftarrow \text{ distributing}$

$\Rightarrow x y = 1 + 10 y$

$\Rightarrow x = \frac{1 + 10 y}{y}$

$\text{the denominator} \ne 0$

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

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