Domain of the $f(x) = \sqrt{x}$
So typically, when introducing functions to students, teachers will say that the domain of this function is $[0,\infty)$.
However, in the curriculum I'm following, one of the first things we did was introduce complex numbers. Thus, shouldn't I be teaching that the domain of this function is all real numbers?
Maybe if we were to specify that the codomain is real numbers then I could understand having the domain be $[0,\infty)$, but we never make such a restriction.
Advice appreciated!
The domain is part of the data of the function. If you have two functions given by the same formula on their respective domains, but with different domains, then they are technically different functions.
Also, the complex square root isn't a function at all in the strict sense—it's a multivalued function, that is, it maps each complex number to a set of complex numbers. (There's no way to assign a single square root to each complex number in a way that yields a continuous function.) So the nonnegative real square root and the multivalued complex square root are just two different things, even if conceptually they're closely related (and they're useful in different contexts).