$f(O(x))$ vs. $O(f(x))$

Question

I understand what $O(f(x))$ means... $O(f(x))=\{g(x): g(x) ≤ cf(x), c,x_0>0 \}$. But I can't find a clear explanation on the meaning of: $f(O(x)).$ Can you explain the exact definition of this? When does $f(O(x))=O(f(x))$? Can you provide examples of when this holds and say why?

Answer 1

As $O(x)$ is set of functions, then $f(O(x))=\{f\circ g \colon g \in O(x) \}$, when it have sense.

Simply example of $f$, without any requirement, to hold $f(O(x))=O(f(x))$ can be identity function i.e. $f(x)=\text{id}(x)=x$.

Let me mention, that one of the well known formulas of mathematical analysis is $$O(O(f(x))) = O(f(x))$$

Where assumed $f(x)>0$.