I understand that the domain is typically defined as the set of objects for which a function is defined. So, given a function $f(x) = x$, how can I figure out its domain? Is $bananas$ part of the domain, given that the function seems to defined for $bananas$ in that $f(bananas) = bananas$? Indeed, is the domain simply everything for this function?
EDIT
I am told that I should specify the domain and codomain as part of the definition of a function, and use something like $f: A \to B : x \mapsto f(x)$. So is $A$ here the domain and $B$ the codomain?
Also, can I say $f : \mathbb{R}\setminus{\{0,1\}} \to \mathbb{R} : x \mapsto \frac{1}{x}$?
When a person is to talk about a function "legally" then he should specify the domain, the codomain, and the corresponding rule of the function. From example, it is sloppy to write "the function $f(x) = x^{2}$" (though of course it would be okay if the context is clear enough); ideally the author might try to say instead "the function $f(x) = x^{2}$ from $\mathbb{R}$ to $\mathbb{R}$" or "the function $f: x \mapsto x^{2}: \mathbb{R} \to \mathbb{R}$". What we are given is merely the corresponding rule "$f(x) = x$" of a mysterious function $f$, so there is nothing much to say from there on.