Example 13. There are $n$ parabolas on a plane, where each two intersect at only two points, and no three intersect at the same point. Find how many parts these $n$ parabolas divide the plane into.
Because the number of parts into which the plane is divided is related to the number of parabolas, we can conjecture that they have a functional relationship, and then attempt to construct this function.

Solution If there are already $n-1$ parabolas satisfying the conditions on the plane, dividing it into $f(n-1)$ parts, and we add the $n$-th parabola (still satisfying the known conditions), then the $n$-th parabola intersects with the previous $n-1$ parabolas at $2(n-1)$ points, dividing the added parabola into $2(n-1)+1=2 n-1$ segments, increasing the total number of plane parts by $2 n-1$ parts. Thus, the function we construct is
$$
f(n)=f(n-1)+2 n-1 . \quad(n \in N)
$$

Using the constructed function recursively, it is easy to find that
$$
f(n)=n^{2}+1 \text {. }
$$