Let $X$ be an $n$-connected elliptic space and $Y$ a non rationally contractible, finite-type, $q$-dimensional CW complex, where $q\leq n$. We show that the function space $X^{Y}$ of free maps from $Y$ into $X$ is formal if and only if the rational cohomology algebra $H^*(X;Q)$ is free, that is, $X$ has the rational homotopy type of a product of odd dimensional spheres.