# What is the domain of  f (x) = 1 / sqrt((2 - x)(6 + x))?

Sep 17, 2015

$x \in \left(- 6 , 2\right)$

#### Explanation:

To be able to calculate $f \left(x\right)$, we have to avoid dividing by 0 and to calculate square root of negative numbers. So,

$\left(\sqrt{\left(2 - x\right) \left(6 + x\right)} \ne 0 \wedge \left(2 - x\right) \left(6 + x\right) \ge 0\right) \iff$
$\left(2 - x\right) \left(6 + x\right) > 0 \iff$
$\left(2 - x > 0 \wedge 6 + x > 0\right) \vee \left(2 - x < 0 \wedge 6 + x < 0\right) \iff$
$\left(x < 2 \wedge x > - 6\right) \vee \left(x > 2 \wedge x < - 6\right) \iff$
$x \in \left(- 6 , 2\right) \vee x \in \emptyset \iff$
$x \in \left(- 6 , 2\right)$