Does infinitely many functions f(n) satisfy Θ(g(n))?

Question

In the context of asymptotic notation, are there infinitely many functions $f(n)$ that is a member of $\Theta( g(n) )$?

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My thoughts

Let's say we set f(n) to be equal to a constant.

We would then always be able to increase either $c_1$ or $c_2$ to be able to create more $f(n)$.

So in the case of constants, I believe that infinitely many $f(n)$ is a member of $\Theta(g(n))$.

Now the question is, does this hold true for other higher order functions? I am not sure..

Answer 1

Yes, infinitely many functions $f(n)$ are $\Theta(g(n))$. For any $g(n)$, the functions given by $$ f(n) = C g(n), $$ where $C > 0$ is a constant, form an infinite family of functions that are $\Theta(g(n))$.