Question

I have two functions:

• f(n) = 2;
• g(n) = 10 ^ 100;

I have to justify if f(n) = BigTheta(g(n)) or not.

My guess is that f(n) is BigTheta(g(n)), since both functions are constants (wich means the functions are proportional), but a my teacher insists that I am wrong.

Am I right? Is there any way I could rest my case?
Sorry if this sounds like a noob question! Thanks.

Answer 1

f(n) <= g(n) * 1
2    <= 10^100     for all n >= 0

Thus f(n) = O(g(n)).

f(n) >= g(n) * 2/(10^100)
2    >= 10^100 * 2/(10^100) = 2      for all n >= 0

And so f(n) = Ω(g(n)).

Both f(n)=O(g(n)) and f(n)=Ω(g(n)) imply f(n) = Θ(g(n)). Yes, you are right.