What is the Domain of $f(x) = x^{1/x}$ looking on wolphram alpha for the function
$$f(x) = x^{1/x}$$
it says that the domain is
for each $x&gt;0$ but for example if I make $x=-3$ it becomes
$$   (-3)^{1/(-3)} = \left(-\frac{1}{3}\right)^{1/3} = -0.69336....  $$
So where am I wrong?
As a real function, $x^y$ is defined as
$$x^y\overset{\text{def}}{=}\mathrm e^{y\ln x}, \enspace\text{ so }\quad x^{\tfrac 1x}=
\mathrm e^{\tfrac{\ln x}x},$$
which requires $x&gt;0$ to be defined.
Its
$$(-3)^{1/(-3)}$$
not
$$-(3^{1/(-3)})$$