1988 AHSME Problem 30

Let f(x) = 4x - x^{2}. Give x_{0}, consider the sequence defined by x_{n} = f(x_{n-1}) for all n \ge 1. For how many real numbers x_{0} will the sequence x_{0}, x_{1}, x_{2}, \ldots take on only a finite number of different values?

\begin{array}{l} \textbf{(A)}\ \text{0}\qquad \textbf{(B)}\ \text{1 or 2}\qquad \textbf{(C)}\ \text{3, 4, 5 or 6}\qquad\\ \textbf{(D)}\ \text{more than 6 but finitely many}\qquad\\ \textbf{(E) }\infty \end{array}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.