Example 6 Try to find $f^{(n)}(x)$ using the fixed point method, where $f(x)=a x+b$.

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Example 6 Try to use the fixed point method to find $f^{(n)}(x)$, where $f(x)=a x+b$.

Solve: Let $f(x)=x$, we get the unique fixed point of $f(x)$ as $x_{0}=\frac{b}{1-a}$. Then $f(x)=a\left(x-x_{0}\right)+f\left(x_{0}\right)$.
$$
\begin{aligned}
f^{(2)}(x) & =a f(x)+b=a^{2}\left(x-x_{0}\right)+a f\left(x_{0}\right)+b \\
& =a^{2}\left(x-x_{0}\right)+a x_{0}+b=a^{2}\left(x-x_{0}\right)+f\left(x_{0}\right), \cdots \\
f^{(n)}(x) & =a^{\prime \prime}\left(x-x_{0}\right)+f\left(x_{01}\right)=a^{n}\left(x-\frac{b}{1-a}\right)+\frac{b}{1-a}
\end{aligned}
$$