Example 4 Determine all polynomials $f(x)$ that satisfy the following condition:
$$
f(x+1) \geqslant \frac{1}{2} f[f(x)]+\frac{3}{2}
$$
If $f(x)$ is an $n$-degree polynomial function, then $f[f(x)]$ is an $n^{2}$-degree polynomial function. Given $n^{2} \leqslant n$, it follows that $f(x)$ is a first-degree polynomial function or a constant function.
Let $f(x)=a x+b$, substituting into (1) we get
$$
a(x+1)+b \geqslant \frac{1}{2}[a(a x+b)+b]+\frac{3}{2},
$$

which simplifies to $a x+a+b \geqslant \frac{1}{2} a^{2} x+\frac{1}{2}(a+1) b+\frac{3}{2}$. This must hold for all $x \in \mathbf{R}$.
This is equivalent to $\left\{\begin{array}{l}a=\frac{1}{2} a^{2} \\ a+b \geqslant \frac{1}{2}(a+1) b+\frac{3}{2}\end{array},\left\{\begin{array}{l}a=2 \\ b \leqslant 1\end{array}\right.\right.$ or $\left\{\begin{array}{l}a=0 \\ b \geqslant 3\end{array}\right.$.
Therefore,
the desired functions are $f(x)=b, \forall b \geqslant 3$ or $f(x)=2 x+b, \forall b \leqslant 1$.