Adding two big Os

Question

If a function A calls n^c functions B that runs in O(n^2) time, what is the time complexity of function A? Is it just polynomial (n^c) as well as c has just gotten bigger?

Answer 1

If a function A calls another function B, the total complexity is the product of the complexities of A and B. So in this case the total complexity is O(n^c · n²) = O(n^{c + 2}).

The general rules for products:

ƒ₁ ∈ O(g₁) and ƒ₂ ∈ O(g₂) ⟹ ƒ₁·ƒ₂ ∈ O(g₁·g₁)

ƒ·O(g) ∈ O(ƒ·g)