Hi, I have this question

Let f: $\displaystyle R \rightarrow R$ (R is the set of real numbers) and let $\displaystyle f^2 = f(f(x))$ intersect the graph of the identity mapping id in just one point (call it p). How many points of period-1 does f have? In how many points does the graph of f intersect that of id?

I know that the point p is a fixed point of $\displaystyle f^2$ and I know that if f has a fixed point then it is also a fixed point of $\displaystyle f^2$, so f can have at most one fixed point. But I just can't seem to get my head around this question.

Please help

Katy