Let $(\mathcal{ M}, c_k, n_k,\kappa)$ be a class of homogeneous Moran sets. Suppose $f(x,y)\in C^3$ is a function defined on $\mathbb{ R}^2$. Given $E_1, E_2\in(\mathcal{ M}, c_k, n_k,\kappa) $, in this paper, we prove, under some checkable conditions on the partial derivatives of $f(x,y)$, that $$f(E_1,E_2)=\{ f(x,y):x\in E_1,y\in E_2\}$$ is exactly a closed interval or a union of finitely many closed intervals. Similar results for the homogeneous self-similar sets with arbitrary overlaps can be obtained. Further generalization is available for some inhomogeneous self-similar sets if we utilize the approximation theorem.
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