1980 AHSME Problem 14

If the function f is defined by f(x)=\frac{cx}{2x+3} ,\quad x\neq -\frac{3}{2} , satisfies x=f(f(x)) for all real numbers x except -\frac{3}{2}, then c is

\text{(A)} \ -3 \qquad \text{(B)} \ - \frac{3}{2} \qquad \text{(C)} \ \frac{3}{2} \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \text{not uniquely determined}

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