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Giving to note the graphs of:

$$y = \sqrt{x}$$

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and of:

$$y = \cos x$$

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there is a way to draw by hand the graph of:

$$y = \sqrt{\cos x}$$

??

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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!

πρόσεχε
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    Looks you're trying to find a pattern between graphs of $f(x)$ and $f(g(x))$. First, maybe start with a small example. Does the graph of $y=x^2$ look anything like $y=x$ ? – AgentS Jun 23 '18 at 20:06
  • Drawing a graph of a function $f$ is usually via points $x \mapsto \langle x,f(x) \rangle$. – tp1 Jun 23 '18 at 20:07

1 Answers1

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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.

Rhys Hughes
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