AMC 12 2005a Problem 20

For each $x$ in $[0,1]$, define

$$f(x) = \begin{cases} 2x, \mathrm{if} 0 \leq x \leq \frac{1}{2}\\ 2-2x, \mathrm{if} \frac{1}{2} < x \leq 1. \end{cases}$$

Let $f^{[2]}(x) = f(f(x))$, and $f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $n \geq 2$. For how many values of $x$ in $[0,1]$ is $f^{[2005]}(x) = 1/2$?

$\textbf{(A) } 0 \textbf{(B) } 2005 \textbf{(C) } 4010 \textbf{(D) } 2005^2 \textbf{(E) } 2^{2005}$