Ygtjki

$sin^2(x)$ has period $pi$ but it seems to me $sqrtsin x$ is not periodic since inside square root has to be positive and when it is negative, it is not defined.

Does it creates problem for periodicity? Can we say square root of the periodic function need to be periodic? Thanks for your help.

This is graph of the $sqrtsin x$ above.

The function $sqrtsin x$ is periodic. If you want (but don't do it in public) think of "undefined" as a real number. Then if $sin x$ is negative, you have $$sqrtsin x = mbox undefined =sqrtsin(x+2pi).$$

Or you can extend to the complex numbers and you'll have periodicity everywhere.

$f$ is periodic, if there is a positive real number $p $ (the period) such that for every $x$ from the domain of $f$ the value of $x$ is the same as value of $(x+p)$ - more formal and more exactly:

$$f text is periodic iffexists p>0: forall x in D(f): x+p in D(f) land f(x) = f(x+p)$$

For the function $g = sqrt f$ of a periodic function $f$ obviously $$x in D(g) iff x + k in D(g)$$

(because $x in D(g) iff f(x) ge 0 iff f(x+k)=f(x) ge 0 $)

and

$$f(x) = f(x+p),$$

so the square root of a periodic function is a periodic one, too.

If a function $f$ is periodic, then there exists $kne0$ such that $f(x)=f(x+k)$ for all $xinmathbbR$. So if $g$ is its square root, we have
$$g(x)=sqrtf(x)=sqrtf(x+k)=g(x+k).$$
And, hence, $g$ is also periodic. BUT note that the square root is only defined in the intervals in which $f$ is nonnegative. Out of them, $g$ is not periodic because it is not even defined.