3. $f(x)=2|x-1 / 2|-1, f^{(n)}(x)=f(f(\ldots f(x) \ldots))$ ( $f$ is repeated $n$ times). Let $L-$ be the broken line which is the graph of the function $f^{(2009)}(x)$. How many segments does the broken line $L$ have?
3. Answer: $2^{2009}$.

The graph of the function $f(x)$ represents a "tick" with a minimum at the point $\left(\frac{1}{2} ; -1\right)$, passing through the points $(-1,2)$ and $(2,2)$. Let's consider what happens to the graph when it is substituted into the function $f$. First, the entire graph is shifted down by $\frac{1}{2}$, then the part of the graph that ends up below the x-axis is flipped, then the graph is stretched by a factor of two and shifted down by 1. During this transformation, each "tick" located between the lines $y=-1$ and $y=2$ turns into two similar but narrower ticks, with the points having ordinates of 2 remaining in place, the minima (which had ordinates of -1) turning into maxima with ordinates of 2, and the points with ordinates of $\frac{1}{2}$ turning into new minima. Thus, each application of the function $f$ doubles the number of segments of the broken line. The first function has 2 segments, so the 2009th function has $2^{2009}$ segments.