# How do you find the domain and range of (3x+2)/(4x-5)?

Aug 25, 2017

$x \in \mathbb{R} , x \ne \frac{5}{4}$
$y \in \mathbb{R} , y \ne \frac{3}{4}$

#### Explanation:

$\text{for } y = \frac{3 x + 2}{4 x - 5}$

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

$\text{solve "4x-5=0rArrx=5/4larrcolor(red)" excluded value}$

$\Rightarrow \text{domain is } x \in \mathbb{R} , x \ne \frac{5}{4}$

$\text{to find any excluded values in the range rearrange making}$
$\text{x the subject}$

$\Rightarrow y \left(4 x - 5\right) = 3 x + 2 \leftarrow \textcolor{b l u e}{\text{ cross-multiplying}}$

$\Rightarrow 4 x y - 5 y = 3 x + 2$

$\Rightarrow 4 x y - 3 x = 2 + 5 y \leftarrow \textcolor{b l u e}{\text{ collect terms in x}}$

$\Rightarrow x \left(4 y - 3\right) = 2 + 5 y \leftarrow \textcolor{b l u e}{\text{ factor out x}}$

$\Rightarrow x = \frac{2 + 5 y}{4 y - 3}$

$\text{the denominator cannot equal zero}$

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

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