Proving that a function f(n) belongs to a Big-Theta(g(n))

Question

Its a exercise that ask to indicate the class Big-Theta(g(n)) the functions belongs to and to prove the assertion.

In this case f(n) = (n^2+1)^10

By definition f(n) E Big-Theta(g(n)) <=> c1*g(n) < f(n) < c2*g(n), where c1 and c2 are two constants.

I know that for this specific f(n) the Big-Theta is g(n^20) but I don't know who to prove it properly. I guess I need to manipulate this inequality but I don't know how

Answer 1

A function f(x) is Θ(g(x)), iff:

• f(x) is O(g(x)), and
• g(x) is O(f(x))

So, while you could try to prove it in a single inequality, I suggest you break it down into those two parts; first prove that for some n>n₀ f(n) < c₁ g(n), and then prove that for some N > N₀ g(N) < c₂ f(N). Once you have proven both parts, separately, you can go back to the definition of Θ to prove that f = Θ(g).

Answer 2

I'm not really an expert at this, but couldn't you prove that f(n) E Ο(n) and that f(n) E Ω(n) and then argue that f(n) E Θ(n) due to the definition of intersection?