# How do you find the domain and range of f(x)= (x+7)/(2x-8)?

Jul 24, 2018

$x \in \mathbb{R} , x \ne 4 , y \in \mathbb{R} , y \ne \frac{1}{2}$

#### Explanation:

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

$\text{solve "2x-8=0rArrx=4larrcolor(red)"excluded value}$

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

$\left(- \infty , 4\right) \cup \left(4 , + \infty\right) \leftarrow \textcolor{b l u e}{\text{in interval notation}}$

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

$\text{let } y = \frac{x + 7}{2 x - 8}$

$y \left(2 x - 8\right) = x + 7$

$2 x y - 8 y = x + 7$

$2 x y - x = 7 + 8 y$

$x \left(2 y - 1\right) = 7 + 8 y$

$x = \frac{7 + 8 y}{2 y - 1}$

$\text{solve "2y-1=0rArry=1/2larrcolor(red)"excluded value}$

$\text{range is } y \in \mathbb{R} , y \ne \frac{1}{2}$

$\left(- \infty , \frac{1}{2}\right) \cup \left(\frac{1}{2} , + \infty\right)$
graph{(x+7)/(2x-8) [-10, 10, -5, 5]}