Edit: Man, I actually had a bachelor's completed when I asked this dumb question, like even more than the terry tao dumb thing. Don't judge me! But to be fair even when I was in calculus we're always asked to 'find the domain' of single variable functions or 'sketch the domain' bivariate functions.
Suppose we have a function, say, $f(x) = x+2$. Its domain is $\mathbb{R}$. How do you prove this? Or is this something not needed to be proven since it is "defined" $\forall$ x $\in \mathbb{R}$?
If to be proven (ignore if not needed): Induction seems to do the trick but that would only cover positive integers. I guess I could cover negative integers using a similar argument. Maybe I could even extend to all rational numbers. What about irrational numbers then?
If not to be proven (ignore if not needed): So highschool teachers should say the domain of $f(x) = x+2$ is $\mathbb{R}$ by definition?
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$\begingroup$You do not and can not prove the domain of a function, you specify it (either explicitly or implicitly) for the function you're discussing.
The function $f(x)=x+2$ is defined for all $x \in \Bbb R$ and hence its domain may be $\Bbb R$. However, you may also define the function $g(x) = x+2$ for $g: [0,1] \to [2,3]$ and then the domain of $g$ is simply $[0,1]$, even though it can be extended to $\Bbb R$.
Also, note that the function $f$ is also defined for all $x \in \Bbb C$ and hence its domain could also be said to be $\Bbb C$.
$\endgroup$ 9 $\begingroup$Yes, absolutely, it should be defined. I had the same experience with my high-school teacher who actually got mad. I do not remember exactly, but apparently it is assumed (without stating) that the domain is the largest possible set on which the (rational, real, complex?) function is defined. I mean, something like $\sqrt{x-4}$ could als be defined on $[2014,\infty)$. So always ask for a specification of the domain.
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