Finite mixtures are just convex combinations of two or more component distributions. Given weights $w_1, \dots, w_n$ and component distributions $P_i$ or densities $p_i$, the distribution and density functions are easy to compute, $$ F(x) = \sum_{i=1}^n w_i P_i(x) $$ $$ f(x) = \sum_{i=1}^n w_i p_i(x) $$ but the inverse distribution $F^{-1}(p)$ or quantile function is not so straightforward.