Let M be a metrizable group. Let G be a dense subgroup of $M^X$. We prove that if G is domain representable, then $G = M^X$. The following corollaries answer open questions. If X is completely regular and $C_p(X)$ is domain representable, then X is discrete. If X is zero-dimensional, T₂, and $C_p(X,𝔻)$ is subcompact, then X is discrete.