Question

Find the domain of the following function:
$$
f(x)=\frac{1}{(1-x) \sqrt{5-x^{2}}}
$$

Answer 1

We know that $\frac{n}{0}$ is undefined for all $n \in \mathbb{R}$ and $\sqrt{x}$ is only defined for $x \geq 0$. The first condition applies to the first term in the denominator and both conditions apply to the second, giving us
$$
(1-x) \neq 0 \text { and } 5-x^{2}>0
$$
The first condition implies $x \neq 1$ while the second implies $|x|<\sqrt{5}$. Putting these together, we find that the domain is
$$
\{x|x \neq 1,| x \mid<\sqrt{5}\} \text { or }(-\sqrt{5}, 1) \cup(1, \sqrt{5})
$$