Example 2 If the linear function $f(x)$ satisfies $f(f(x))=4x+1$, then $f(x)$
Let $f(x)=a x+b$, then
$$
f(f(x))=a(a x+b)+b=4 x+1,
$$

Thus $\left\{\begin{array}{l}a^{2}=4, \\ a b+b=1,\end{array}\right.$ solving this yields $\left\{\begin{array}{l}a=2, \\ b=\frac{1}{3}\end{array}\right.$ or $\left\{\begin{array}{l}a=-2, \\ b=-1,\end{array}\right.$ Therefore,
$$
f(x)=2 x+\frac{1}{3} \text { or } f(x)=-2 x-1 .
$$

Key Points for Solving The problem We refer to $f(g(x))$ as the composite function of functions $f(x)$ and $g(x)$. $f(f(x))$ is a special type of composite function, being the composition of function $f(x)$ with itself.