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 $f: \mathbb{R} \rightarrow \mathbb{R}$?

OK. Yea, $f: \mathbb{R} \rightarrow \mathbb{R}$ where $D_f=\mathbb{R}\subseteq\mathbb{R}$ and $R_f=(0,1]\subseteq\mathbb{R}$. 
