Example 3. Find the inverse of the function $y=\frac{1}{2} \log _{2} \frac{1+x}{1-x}$
Solve for domain, range: From $\frac{1+x}{1-x}>0$, we get $-1<x<1$.
Inverse solution: From $y=\frac{1}{2} \log _{2} \frac{1+x}{1-\frac{x}{x}}$, we get
$$
\begin{array}{l}
2^{2 y}=\begin{array}{c}
1+x \\
1-x
\end{array}, \\
\therefore \quad x=\frac{2^{y}-2^{-y}}{2^{y}+2^{-y}} \text {. } \\
\end{array}
$$

Interchange: $y=\frac{2^{x}-2^{-x}}{2^{x}+2^{-x}}, x \in R$, is the required inverse function.