About $f(x) = \frac{1}{1+x^{2}}$ I realize that this function has a horizontal asymptote $y=0$.  And that the range of this function is $(0, 1]$ 
Is the function $f: \mathbb{R} \rightarrow \mathbb{R}$ since for every $x \in \mathbb{R}$  $\exists$ a $f(x) \in \mathbb{R}$.
i.e. can I say $f: \mathbb{R} \rightarrow \mathbb{R}$?

Yes, it's ok to say that. Technically, the definition of a function includes a description of its domain, so you are right to wonder. The second $\mathbb{R}$ is fine in any case, since it represents the codomain and not the range.
