# How do you find the domain and range of f(x) = 7 / (2x-10)?

Apr 7, 2017

$x \in \mathbb{R} , x \ne 5$
$y \in \mathbb{R} , y \ne 0$

#### 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.

$\text{solve " 2x-10=0rArrx=5larrcolor(red)" value that x cannot be}$

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

$\text{Rearrange f(x) making x the subject}$

$y = \frac{7}{2 x - 10}$

$\Rightarrow y \left(2 x - 10\right) = 7$

$\Rightarrow 2 x y - 10 y = 7$

$\Rightarrow 2 x y = 7 + 10 y$

$\Rightarrow x = \frac{7 + 10 y}{2 y}$

$\text{Apply the same reasoning as for domain}$

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