Example 3 Given $f(2 x+1)=2 x^{2}+1$, then $f(x)=$
Let $t=2x+1$, then $x=\frac{1}{2}(t-1)$, thus
$$
f(t)=2\left[\frac{1}{2}(t-1)\right]^{2}+1=\frac{1}{2}(t-1)^{2}+1,
$$

Therefore, $f(x)=\frac{1}{2}(x-1)^{2}+1$.
Solution Guide: Using the substitution $t=g(x)$ to solve for $x=p(t)$, and then substituting it into the expression of the function $f(g(x))$, is an important method to solve the problem of "given the expression of the composite function $f(g(x))$, find the expression of the function $f(x)$".