The domain of $f(x)=x^{1/x}$ I've been taught that the domain of the function $f(x) = x^{1/x}$ is $x > 0$ assuming the function to be from $\mathbb{R} \to \mathbb{R}$. But $f(-1) = (-1)^{(-1)} = -1$, so why does the domain not include $x = -1$?

Generally, in calculus, you consider expressions of the form
$$f(x)^{g(x)}$$
as an abbreviation for
$$\mathrm e^{g(x)\ln f(x)},$$
which is only defined when $f(x)>0$. (Do you see why both expressions are equal in that case?)
You are correct to say that this does not tell the whole truth, but the missing cases are generally not interesting from the point of view of calculus.
