17. Let $f(x)=|1-2 x|, x \in[0,1]$, and denote $f_{1}(x)=f(x), f_{2}(x)=f\left(f_{1}(x)\right), f_{3}(x)=$ $f\left(f_{2}(x)\right), \cdots, f_{n+1}(x)=f\left(f_{n}(x)\right)$, try to find how many roots the equation $f_{n}(x)=\frac{1}{2} x$ has in $[0,1]$.
17. Since the graph of the function $f_{1}(x)=f(x)=2\left|\frac{1}{2}-x\right|$ is symmetric about the line $x=\frac{1}{2}$, $(x \in[0,1])$

Therefore, the graph of $f_{2}(x)=2^{2}\left|\frac{1}{2^{2}}-\right| \frac{1}{2}-x||$ is first symmetric about $x=\frac{1}{2}$, and when $0 \leqslant x \leqslant \frac{1}{2}$, $f_{2}(x)=2^{2}\left|x-\frac{1}{4}\right|$, its graph is also symmetric about $x=\frac{1}{4}$. Thus, it is easy to know that $y=f_{2}(x)$ is periodic with $T=\frac{1}{2}$ on $[0,1]$. As shown in Figure 4-3, the equation $f_{2}(x)=\frac{1}{2} x$ has $2^{2}$ roots.

Similarly, the graph of $y=f_{x}(x)$ is first symmetric about $x=\frac{1}{2}(x \in(0,1))$, and also symmetric about $x=\frac{1}{4}\left(x \in\left(0, \frac{1}{2}\right)\right)$, and further symmetric about $x=\frac{1}{8}, x \in\left[0, \frac{1}{4}\right]$. Therefore, $y=f_{x}(x)$ is periodic with $T=\frac{1}{4}$ on $[0,1]$. From the above discussion, the graph of $y=f_{3}(x)$ is as shown in Figure 4-4, so the equation $f_{3}(x)=\frac{1}{2} x$ has $2^{3}$ roots.

In general, the function $y=f_{n}(x)$ is symmetric about the axes $x=\frac{1}{2}, x=\frac{1}{4}, x=\frac{1}{8}, \cdots, x=\frac{1}{2^{n-1}}$ in the intervals $[0,1],\left[0, \frac{1}{2}\right], \left[0, \frac{1}{4}\right], \cdots,\left[0, \frac{1}{2^{n-1}}\right]$, and the points where the graph intersects the x-axis are exactly the intersections of these axes with the x-axis. Hence, $y=f_{n}(x)$ is periodic with $T=\frac{1}{2^{n-1}}$ on $[0,1]$. Therefore, based on the periodicity, the graph of $y=f_{n}(x)$ can be drawn (figure omitted), consisting of $2^{n-1}$ "V" shapes connected together. According to the graph, the equation $f_{n}(x)=\frac{1}{2} x$ has $2^{n}$ roots on $[0,1]$.

Note: Some mathematical problems may seem unrelated to function properties on the surface, but they actually involve some properties of function graphs, especially periodicity. To understand the graph properties and grasp the essence of the problem, it is necessary to use the periodicity of the function to assist.