Example 10 (1) Find the domain of the function $y=\frac{\sqrt{x^{2}-4}}{\log _{2}\left(x^{2}+2 x-3\right)}$;
(2) Given that the domain of the function $f(x)$ is $[-1,1]$. Find the domain of $f(a x)+f\left(\frac{x}{a}\right)$, where
(1) The domain of the function is the solution set that satisfies the following conditions.
$$
\left\{\begin{array}{l}
x^{2}-4 \geqslant 0, \\
x^{2}+2 x-3>0, \\
x^{2}+2 x-3 \neq 1 .
\end{array}\right.
$$

Therefore, the domain is $(-\infty,-1-\sqrt{5}) \cup(-1-\sqrt{5},-3) \cup[2,+\infty)$.
(2) The domain of $f(a x)+f\left(\frac{x}{a}\right)$ is the intersection of the following two sets:
$$
\begin{array}{l}
D_{1}=\{x \mid-1 \leqslant a x \leqslant 1\}=\left[-\frac{1}{a}, \frac{1}{a}\right], \\
D_{2}=\left\{x \left\lvert\,-1 \leqslant \frac{x}{a} \leqslant 1\right.\right\}=[-a, a] .
\end{array}
$$

When $a \geqslant 1$, $a \geqslant \frac{1}{a},-a \leqslant-\frac{1}{a}$, hence $D_{1} \cap D_{2}=D_{1}$;
When $0<a<1$, $a<\frac{1}{a},-\frac{1}{a}<-a$, hence $D_{1} \cap D_{2}=D_{2}$.
Therefore, the domain of $f(a x)+f\left(\frac{x}{a}\right)$ is
$$
[-a, a] \text { (when } 0<a<1) \text { or }\left[-\frac{1}{a}, \frac{1}{a}\right] \text { (when } a \geqslant 1 \text {. }
$$