Finding the Range of a Function, Algebraically

Let $A \subseteq \mathbb{R}$. Given $A$, is it always possible to construct a (not necessarily continuous) function $f: \mathbb{R} \to \mathbb{R}$, where $f(x)$ is defined for all $x$, such that the range of $f$ is $A$? If not, give a counterexample.