Minimal conditions for $\widetilde{d}$ to be metric

Question

Let $(X,d)$ be an arbitrary metric space and $f:[0,\infty) \rightarrow [0,\infty)$

What are the minimal conditions for function $f$ in order

$\widetilde{d} = f \circ d: X \times X \rightarrow [0,\infty) $ be also metric on $X$?

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Do we need to prove that $\widetilde{d} = f(d(x,y))$, for all $x,y \in X$ satisfies the definition of metric on $X$?

Please help!

Answer 1

Two things:

a. $f(x)=0$ iff $x=0$.

b. $f(x+y)\le f(x)+f(y)$.

They are necessary and sufficient conditions.