In a textbook, I came across a definition of big-oh notation, it goes as follows:
We say that $f(x)$ is $O(g(x))$ if there are constants $C$ and $k$ such that $$|f(x)| \le C|g(x)|$$ whenever $x \gt k$.
If I'm not mistaken, this basically translates to: $$f(x) = O(g(x)) \Leftarrow\Rightarrow (\exists C,\exists k|C,k \epsilon \Bbb R \land (x \gt k \rightarrow |f(x)| \le C|g(x)|))$$ Now, I have two questions regarding this statement:
Is my verbose translation correct?
What exactly does this definition of big-oh notation mean about the usage of big-oh notation, because from what I understand through computer science, big-oh is used to represent the computational complexity of an algorithm. So how does this relate to the complexity of an algorithm (if at all)?