Number of intersection point of $\cos^{-1}$ function

Question

If $f:[0,4\pi]\to[0,\pi]$ is the function defined by $f(x)=\cos^{-1}(\cos x)$, the number of points $x\in[0,4\pi]$ satisfying the equation $f(x)=\frac{10-x}{10}$ is ______.

I am mapping the curve $f(x)=\cos^{-1}(\cos x)$. It is equivalent to $y=x$, in that case answer is '$1$', but actual answer is $3$. Where am I making mistake?

Answer 1

$$\cos^{-1}(\cos x)=\begin{cases} x&\mbox{if } 0\le x\le\pi \\2\pi-x & \mbox{if }\pi<x\le2\pi\\x-2\pi & \mbox{if }2\pi<x\le3\pi\\4\pi-x & \mbox{if }3\pi<x\le4\pi\end{cases}$$

Answer 2

Hint: Remember that $\cos^{-1}(x)$ is always defined to be the angle $\theta$ between $0$ and $\pi$ with $\cos\theta = x$. In particular, if $x$ is outside $[0, \pi]$, then $\cos^{-1}(\cos(x)) \neq x$.