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)?

Let $X,Y$ be metric space and suppose that $f:X\rightarrow Y$ is a uniform embedding; i.e.: $$ \omega(d_X(x,z))\leq d_Y(f(x),f(z)) \leq \Omega(d_X(x,z)), $$ where $\omega\leq \Omega$ are both strictly increasing continuous functions mapping $[0,\infty)$ to itself and which fix $0$.

Is $f$ a quasisymmetry? I.e.: does there exist a monotone function $\eta:[0,\infty)\rightarrow [0,\infty)$ satisfying $$ \frac{d_Y(f(x),f(y))}{d_{Y}(f(x),f(z))} \leq \eta\left(\frac{d_X(x,y)}{d_X(x,z)}\right) . $$

Note on Edit: I have reduced my previous question down to this more general one; since quasisymmetries preserve the doubling property.

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. Then, is $f(X)$ doubling also (and if so what does its doubling constant look like)?

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)?

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 $f(X)$?

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. Then, is $f(X)$ doubling also (and if so what does its doubling constant look like)?