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?
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2Arguably, $f(O(\cdot))$ is an abuse of notation, as it treats a set as a "generic" representative of that set. There is some discussion in this related question. In what context did you encounter $F(O(x))$? I sometimes see $2^{O(n)}$, but not much in the "generic" form. – Discrete lizard Sep 17 '21 at 09:33
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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 $f(x)>0$.
zkutch
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