Let $M_1$ and $M_2$ be metric spaces with metrics $\rho_1$ and $\rho_2$ respectively. What are some necessary and sufficient conditions on $f:\mathbb{R}_{+}^2\to\mathbb{R}_{+}$ that make $\rho((x_1,x_2),(y_1,y_2))=f(\rho_1(x_1,y_1),\rho_2(x_2,y_2))$ a metric on $M_1\times M_2$? More generally, how can $f$ be characterized? Does this type of function have an existing name?
Edit: In this question, $\mathbb{R}_+$ is the set of nonnegative real numbers.