Example 4 Given the function $f(x)=\log _{a} \frac{x+b}{x-b}(a>1, b>0)$, find
(1) the domain of $f(x)$; (2) the parity of $f(x)$; (3) the monotonicity of $f(x)$; (4) the inverse function of $f(x)$.
(1) $\frac{x+b}{x-b}>0 \Rightarrow x \in(-\infty,-b) \cup(b,+\infty)$.
(2) For any $x \in(-\infty,-b) \cup(b,+\infty)$, we have
$$
\begin{aligned}
f(-x) & =\log _{a} \frac{-x+b}{-x-b}=\log _{a} \frac{-(x-b)}{-(x+b)} \\
& =\log _{a}\left(\frac{x+b}{x-b}\right)^{-1}=-\log _{a}\left(\frac{x+b}{x-b}\right)=-f(x),
\end{aligned}
$$

Therefore, $f(x)$ is an odd function.
(3) Since the function $\mu=\frac{x+b}{x-b}=1+\frac{2 b}{x-b}$ is decreasing on $(-\infty, b)$ and also decreasing on $(b,+\infty)$. Given that the domain of $f(x)=\log _{a} \frac{x+b}{x-b}$ is $(-\infty,-b) \cup(b,+\infty)$ and $a>1$, the function is decreasing on the intervals $(-\infty,-b)$ and $(b,+\infty)$.
(4) $y=\log _{a} \frac{x+b}{x-b} \Leftrightarrow a^{y}=\frac{x+b}{x-b} \Leftrightarrow x=\frac{b\left(a^{y}+1\right)}{a^{y}-1}$, so $f^{-1}(x)=$ $\frac{b\left(a^{x}+1\right)}{a^{x}-1}(x \neq 0)$