# What is the domain and range of  y = (3(x-2))/x?

Jul 11, 2017

$x \in \mathbb{R} , x \ne 0 , y \in \mathbb{R} , y \ne 3$

#### Explanation:

The denominator of y cannot be zero as this would make y undefined.

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

$\text{domain is } x \in \mathbb{R} , x \ne 0$

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

$\Rightarrow x y = 3 x - 6 \leftarrow \textcolor{b l u e}{\text{ cross-multiply}}$

$\Rightarrow x y - 3 x = - 6 \leftarrow \text{ collect terms in x}$

$\Rightarrow x \left(y - 3\right) = - 6 \leftarrow \text{ common factor of x}$

$\Rightarrow x = - \frac{6}{y - 3}$

$\text{the denominator cannot equal zero}$

$y - 3 = 0 \Rightarrow y = 3 \leftarrow \textcolor{red}{\text{ excluded value}}$

$\text{range is } y \in \mathbb{R} , y \ne 3$