Problem Database
Home
Add Problem
Search
Compare
Rankings
Tag Untagged
Edit Problem
Problem Description
For each \( x\) in \( [0,1]\), define \[ f(x)=\begin{cases}2x, &\text { if } 0 \leq x \leq \frac {1}{2}; \\ 2 - 2x, &\text { if } \frac {1}{2} < x \leq 1. \end{cases} \]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 \( f^{[2005]}(x) = \frac {1}{2}\)? $$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2005 \qquad \textbf{(C)}\ 4010 \qquad \textbf{(D)}\ 2005^2 \qquad \textbf{(E)}\ 2^{2005}$$
Diagram (TikZ Code)
Solution
E
Tags
Difficulty
Source
Save Changes