CMOQR 2014 Problem 1

Let $f : \mathbb{Z} \rightarrow \mathbb{Z}^+$ be a function, and define $h : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}^+$ by $h(x, y) = \gcd (f(x), f(y))$ . If $h(x, y)$ is a two-variable polynomial in $x$ and $y$ , prove that it must be constant.