how to draw graph of cos inverse cos x

Here in this post ,we will check out how to draw the graph of cos inverse cos x i.e

$f(x) = cos^{-1} cos (x)$

We know by definition

$f(x) = cos^{-1} cos (x) = x $ if $x \in [0, \pi]$

Lets check out the other values

if $x \in (\pi, 2\pi]$

$\pi < x \leq 2\pi $
$- 2\pi \leq -x < -\pi$
$0 \leq 2\pi -x < 2\pi – \pi$
$0 \leq 2\pi -x < \pi$

Now $cos(2\pi -x) =co x$

So $f(x) = cos^{-1} cos (x) = cos^{-1} cos (2\pi -x)= 2\pi -x$

Therefore the graph will be

Here f(x) is a periodic function with period $2 \pi$, So same graph will be repeated for all the $2 \pi$ intervals