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 $B_X(x,r/2)$.

Suppose that $f:X\rightarrow Y$ is uniformly continuous and that $f|_{f(X)}^{-1}$ is also uniformly continuous on $f(X)$. Then, is $f(X)$ doubling also (and if so what does its doubling constant look like)?