AHSME 1980 Problem 14

If the function $f$ is defined by $$f(x)=\frac{cx}{2x+3} , x\neq -\frac{3}{2} ,$$ satisfies $x=f(f(x))$ for all real numbers $x$ except $-\frac{3}{2}$, then $c$ is $\text{(A)} \ -3 \text{(B)} \ - \frac{3}{2} \text{(C)} \ \frac{3}{2} \text{(D)} \ 3 \text{(E)} \ \text{not uniquely determined}$