Example 7 Given $f(x)=\frac{x+6}{x+2}$, find $f^{(n)}(x)$
Let $\varphi(x)=\frac{\alpha x+\beta}{x+r}, g(x)=a x$, and satisfy $f(x)=\varphi^{-1}(g(\varphi(x)))$ $f(x)=\frac{x+6}{x+2}=x, x=2$ or -3 is a fixed point of $f(x)$. The fixed points of $g(x)=a x$ are 0 and $\infty$. $g(x)$ transforms $2, -3$ into $0, \infty$. We can let $\varphi(x)=\frac{x-2}{x+3}$, then $\varphi^{-1}(x)=\frac{3 x+2}{1-x}$. Thus,
$$
\begin{array}{c}
g(x)=\varphi\left(f\left(\varphi^{-1}(x)\right)\right)=-\frac{x}{4} \\
f^{(n)}(x)=\varphi^{-1}\left(g^{(n)}(\varphi(x))\right)=\frac{\left(2 \cdot(-4)^{n}+3\right) x+6\left((-4)^{n}-1\right)}{\left((-4)^{n}-1\right) x+3 \cdot(-4)^{n}+2}
\end{array}
$$