Let $X,Y$ be metric space with $X$ $C$-doubling; i.e: each $B_X(x,r)$ can be covered by at-most $C$ balls of radius $B_X(x,r/2)$.

What (reasonable) condition does one need on a Lipschtiz map $f:X\rightarrow Y$ to ensure that $f(X)$ is also doubling? In which case, what is the doubling constant of $f(X)$?