4. Find the function $f(x)$ that satisfies the inequality $f[f(x)] \geqslant x^{2}, x \in(1,+\infty)$.
4. Using the method of undetermined functions. Let $f(x)=x^{\alpha}, \alpha \in \mathbf{R}$, then
$$
f[f(x)]=f^{\alpha}(x)=\left(x^{\alpha}\right)^{\alpha}=x^{\alpha^2} \geqslant x^{2}, \alpha^{2} \geqslant 2 .
$$

Therefore, the solution to the given functional inequality is $f(x)=x^{\alpha}, \alpha \in(-\infty,-\sqrt{2}) \cup(\sqrt{2},+\infty)$.