# How do you find the domain of (sqrt(9-x^2)) / (x-1)?

Nov 1, 2015

$x \in \left\{\left[- 3 , 1\right) \cup \left(1 , 3\right]\right\}$

#### Explanation:

The domain is all the possible values of $x$
To find the domain, we must find the values of $x$ for which the function is undefined

The function,

$f \left(x\right) = \frac{\sqrt{9 - {x}^{2}}}{x - 1}$

will be undefined if
[1] The denominator becomes 0
[2] The term inside the square becomes negative

Hence,

$x - 1 \ne 0$
$\implies x \ne 1$

$9 - {x}^{2} > 0$

$- {x}^{2} > - 9$

${x}^{2} \le 9$

$\implies - 3 \le x \le 3$

Hence, the domain is all the real numbers between -3 and 3 except 1