How do you find the domain and range of g(x) = (2x – 3)/( 6x - 12)?

Aug 11, 2017

$x \in \mathbb{R} , x \ne 2$
$y \in \mathbb{R} , y \ne \frac{1}{3}$

Explanation:

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

$\text{solve "6x-12=0rArrx=2larrcolor(red)" excluded value}$

To find any excluded values in the range rearrange g(x) making x the subject.

$g \left(x\right) = y = \frac{2 x - 3}{6 x - 12}$

$\Rightarrow y \left(6 x - 12\right) = 2 x - 3 \leftarrow \textcolor{b l u e}{\text{ cross-multiplying}}$

$\Rightarrow 6 x y - 12 y = 2 x - 3$

$\Rightarrow 6 x y - 2 x = 12 y - 3$

$\Rightarrow x \left(6 y - 2\right) = 12 y - 3$

$\Rightarrow x = \frac{12 y - 3}{6 y - 2}$

$\text{equate denominator to zero for excluded value}$

$6 y - 2 = 0 \Rightarrow y = \frac{1}{3} \leftarrow \textcolor{red}{\text{ excluded value}}$

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

$\text{range is } y \in \mathbb{R} , y \ne \frac{1}{3}$