Algorithm comparison I am learning Big O and Big theta notation and confused the certain case.
I have two functions,
function 1(f1)
$$ n * n^{1/2} $$
function 2(f2)
$$ 1.001^n $$

in smaller cases (10,000) f1 is much faster than f2, however, it is vice versa for bigger case (10,000,000). For the smaller case, I can say
$$ f2 = O(f1) $$
In the bigger case, I can say
$$ f1 = O(f2) $$
and I think this case is,
$$ f1 = \theta(f2) $$ 
(If ^ this is wrong, please let me know)
At this case, can I say ??
$$ f1 \equiv f2 $$
No, you can't say that, because $O$ doesn't mean "For small inputs, [something] happens" and $\Theta$ doesn't mean "For some inputs, one of the functions is bigger and, for other inputs, the other one is." In mathematics, it is essential to use the actual definition of the concepts that you're working with, and to prove that the definition applies in your situation.