Example 1 Given $f_{1}(x)=\frac{2 x-1}{x+1}, f_{n, 1}(x)=f_{1}\left[f_{n}(x)\right]$. Find $f_{2 x}(x)$
$$
\text { Solve } \begin{aligned}
f_{2}(x) & =\frac{[2(2 x-1) /(x+1)]-1}{[(2 x-1) /(x+1)]+1}=\frac{x-1}{x} \\
f_{1}(x) & =\frac{[(x-1) / x]-1}{(x-1) / x}=\frac{1}{1-x}, f_{6}(x)=\frac{[1 /(1-x)-1]}{1 /(1-x)}=x
\end{aligned}
$$

Thus, $f_{2 x}(x)=f_{22}(x)=f_{15}(x)=\cdots=f_{4}(x)=\frac{1}{1-x}$