Squared Square Root confusion I am confused about the following:
$$\sqrt{x^2}=?\ (\sqrt x)^2 =?\ x^{2/2}$$
The source of confusion is: Let
$$f(x)=\sqrt {x^2}$$
$$g(x)=\ (\sqrt x)^2 $$
Then
$$f(-3)=3$$
$$g(-3)=\mbox{undefined}?$$
While the range of $f(x)$ and $g(x)$ might be the same, their domain appears to be different ($g(x)$ restricts $x$ to be positive). At the same time, both can be written as $$x^{2/2}$$
What am I missing?
-Thank you :)

The problem is 
$$\sqrt{x^2}=(\sqrt{x})^2$$
only if $x \ge 0$. If we are talking about real numbers.
In general we can write:
$$\sqrt{x^2}=\left(\sqrt{|x|}\right)^2 \Leftrightarrow \sqrt{x^2}=|x|$$
