# How do you find the domain and range of g(x)=8/(8-3x)?

Feb 2, 2018

$x \in \mathbb{R} , x \ne \frac{8}{3}$
$y \in \mathbb{R} , y \ne 0$

#### 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.

$\text{solve "8-3x=0rArrx=8/3larrcolor(red)"excluded value}$

$\Rightarrow \text{domain is } x \in \mathbb{R} , x \ne \frac{8}{3}$

$\left(- \infty , \frac{8}{3}\right) \cup \left(\frac{8}{3} , + \infty\right) \leftarrow \textcolor{b l u e}{\text{in interval notation}}$

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

$y = \frac{8}{8 - 3 x}$

$\Rightarrow y \left(8 - 3 x\right) = 8$

$\Rightarrow 8 y - 3 x y = 8$

$\Rightarrow - 3 x y = 8 - 8 y$

$\Rightarrow x = \frac{8 - 8 y}{- 3 y}$

$\text{the denominator cannot equal zero}$

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

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

$\left(- \infty , 0\right) \cup \left(0 , + \infty\right) \leftarrow \textcolor{b l u e}{\text{in interval notation}}$
graph{8/(8-3x) [-10, 10, -5, 5]}