Throughout school I thought that the pre-image was a subset of the domain, not that they were necessarily the same. When I spoke of a function f:R->R, I didn't think that this meant that f was defined on all of R, I thought √x was a function from R to R.
Now I am hearing conflicting things, that the domain is actually the exact same thing as the pre-image.
What convention is the norm?
Best Answer
First, a point to clear up a confusion you seems to have, the way function usually get defined through those operation actually make the domain implicit: you take the biggest set such that the expression can make sense. For $\sqrt{x}$ for example, the domain is actually $[0,\infty)$ and not $\mathbb{R}$, so the function is actually $[0,\infty)\rightarrow\mathbb{R}$; for $\frac{1}{x}$ the domain is $\mathbb{R}\backslash\{0\}$ so it is $\mathbb{R}\backslash\{0\}\rightarrow\mathbb{R}$.
When you say "preimage", you need to specify the function and what the preimage is of. For example, if the function is $x^{2}$, then the preimage of $\{1,4\}$ for this function is $\{-2,-1,1,2\}$ which is a proper subset of the domain $\mathbb{R}$. The preimage of the range of the function (not to be confused with the codomain, which is usually just $\mathbb{R}$) is indeed the domain; and the preimage of some proper subset of the range would be a proper subset of the domain.