Q:

Graph of the square root of a function

Giving to note the graphs of:
$$y = \sqrt{x}$$

and of:
$$y = \cos x$$

there is a way to draw by hand the graph of:
$$y = \sqrt{\cos x}$$
??

The only way that is possible for me is the most brutal one: by points. Perhaps there is a more elating way and I can not see it. Thank you!

A:

Notice that  $\cos(x)$ only has values in the range $(-1,1)$, and that for $0<x<1, \sqrt x>x$, and that at $x=0$ and $x=1$ , $\sqrt x =x$.
Additionally, $\cos(x)>0$ in the range $([2k-\frac12]\pi,[2k+\frac12]\pi), k\in \Bbb Z$, and $<0$ otherwise. So to sketch $\sqrt{\cos(x)}$ reasonably accurately, take the graph of $\cos(x)$ when it is positive, and draw an arc that touches it when it equals $0$ (at $x=2k\pi$) or $1$ (at $x=(2k\pm\frac12)\pi$) and is slightly higher elsewhere.

