# How do you find the domain and range of (5x-3) / (2x +1)?

May 31, 2017

$x \in \mathbb{R} , x \ne - \frac{1}{2}$
$y \in \mathbb{R} , y \ne \frac{5}{2}$

#### Explanation:

$\text{the domain is defined for all real values of x except for}$
$\text{values of x which make the denominator equal zero}$

$\text{to find the value that x cannot be, equate the denominator}$
$\text{to zero and solve}$

$\text{solve "2x+1=0rArrx=-1/2larrcolor(red)" excluded value}$

$\Rightarrow \text{domain is } x \in \mathbb{R} , x \ne - \frac{1}{2}$

$\text{to find any excluded values in the range, rearrange}$

$y = \frac{5 x - 3}{2 x + 1} \text{ making x the subject}$

$\Rightarrow y \left(2 x + 1\right) = 5 x - 3 \leftarrow \textcolor{b l u e}{\text{ cross-multiplying}}$

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

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

$\Rightarrow x \left(2 y - 5\right) = - \left(3 + y\right)$

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

$\text{the denominator cannot equal zero}$

$\text{solve "2y-5=0rArry=5/2larrcolor(red)" excluded value}$

$\Rightarrow \text{range is } y \in \mathbb{R} , y \ne \frac{5}{2}$