Example 2. Find the domain of the function
$$
y=\frac{\sqrt{(4-x)(x+3)}\left(\sqrt{x^{2}-1}\right)^{0}}{(x-3) \lg (x+2)}
$$

The domain of this example is selected from the book "Comprehensive Application of Middle School Mathematics" (edited by Wei Jingnong and Zhang Wenxiong, Zhejiang Science and Technology Press). Out
To make the function defined, it is necessary and sufficient that
$$
\begin{array}{l}
\left\{\begin{array}{l}
(4-x)(x+3) \geqslant 0, \\
x^{2}-1>0, \\
(x-3) \lg (x+2) \neq 0, \\
x+2>0
\end{array}\right. \\
\Leftrightarrow\left\{\begin{array}{l}
-3 \leqslant x \leqslant 4, \\
x \neq \pm 1, \\
x \neq -1 \text { and } x \neq 3, \\
x>-2
\end{array}\right.
\end{array}
$$

The solution set: $-2<x<-1$ or $1<x \leqslant 4$ and $x \neq 3$,
The domain is $\{x \mid -2<x<-1$ or $1<x \leqslant 4$ and $x \neq 3\}$.