What is the domain of $f(x)=x^x$? What is the domain of $f(x)=x^x$ ? 
I used Wolfram alpha where it is said that the domain is all positive real numbers. Isn't $(-1)^{(-1)} = -1$ ? Why does the domain not include negative real numbers as well?
I also checked graph and its visible for only $x>0$ . Can someone help me clarify this?

The expression $x^y$ can be assigned a reasonable meaning for all real $x$ and all rational numbers of the form $y=m/n$, where $m$ is even and $n$ is odd and positive. Thus $x^y=(x^m)^{1/n}$, interpreted as the unique real $n$th root of $x^m$ (define $0^0$ to be $1$). Since every real number can be arbitrarily well approximated by such "even/odd" rationals, by continuity, a synthetic definition of $x^y$ can be obtained for all real $x$ and $y$. For example, using this definition, a graph can be plotted for the relation $y^y=x^x$, which runs smoothly as a loop through all four quadrants (along with the obvious line $y=x$).
