Example 2 Find the direct proportion function $f(x)$ that satisfies the inequality $f[f(x)] \geqslant x-3, x \in \mathbf{R}$.
Assume the method of undetermined function. Let $f(x)=a x, a \in \mathbf{R}$, then
$$
f[f(x)]=a f(x)=a^{2} x \geqslant x-3
$$

For $x \in \mathbf{R}$ to always hold, it must be that $a^{2}=1$, so the solution to the given functional inequality is $f(x)=-x$ or $f(x)=x$.