# How do you find the domain and range of y=(3x-2)/(4x+1)?

Jan 16, 2018

$x \in \mathbb{R} , x \ne - \frac{1}{4}$
$y \in \mathbb{R} , y \ne \frac{3}{4}$

#### Explanation:

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

$\text{solve "4x+1=0rArrx=-1/4larrcolor(red)"excluded value}$

$\Rightarrow \text{domain is } x \in \mathbb{R} , x \ne - \frac{1}{4}$

$\text{for the range rearrange making x the subject}$

$\Rightarrow y \left(4 x + 1\right) = 3 x - 2$

$\Rightarrow 4 x y + y = 3 x - 2$

$\Rightarrow 4 x y - 3 x = - 2 - y$

$\Rightarrow x \left(4 y - 3\right) = - 2 - y$

$\Rightarrow x = \frac{- 2 - y}{4 y - 3}$

$\text{solve "4y-3=0rArry=3/4larrcolor(red)"excluded value}$

$\text{range is } y \in \mathbb{R} , y \ne \frac{3}{4}$