If a function $f(x)$ is constant complexity $f(x) = O(1)$ , describe $C$ and $k$.

Question

If a function $f(x)$ is constant complexity $f(x) = O(1)$ , describe $C$ and $k$.

This needs to be described in terms of the relation of $C$ and $k$.

There exists constants $C$ and $k$ such that $C|f(x)| \le |1|$, for all $x > k$.

The above cannot be true because using the explanation from the book of discrete math:

$f(x)$ is $O(g(x))$ if there are constants $C$ and $k$ such that $$\forall x > k, |f(x)| \le C|g(x)|.$$

So if $f(x) = O(1)$, then it would mean $O(1) \le C|g(x)|$ , or $C|g(x)| \ge O(1)$.

Instead the below statement is true.

There exists constants $C$ and $k$ such that $C|f(x)| \ge |1|$ for all $x > k$.

Is this a correct understanding?

Answer 1

I submitted my work and got it wrong, just wanted to share with others who are looking for answers to this quesion. It is :

There exists constants $C$ and $k$ such that $|f(x)| ≤ C$ for all $x > k$.