Let f(n) = n and g(n) = n(1 + sin n), where n is a positive integer. Which of the following statements is/are correct? I. f(n) = O(g(n)) II. f(n) = Ω(g(n))

Let f(n) = n and g(n) = n(1 + sin n), where n is a positive integer. Which of the following statements is/are correct? I. f(n) = O(g(n)) II. f(n) = Ω(g(n)) Correct Answer Neither I nor II

Concept:

Sin function value ranges from -1 to + 1. (-1, 0, 1)

Explanation:

Case 1: when sin(n) is -1,

g(n) = n^{(1 - 1)} = n⁰ = 1

so, for this case f(n) > g(n) i.e. g(n) = O (f(n))

So, statement 1 is incorrect.

Case 2: when sin(n) is +1,

g(n) = n^{(1 + 1)} = n²

so, for this case f(n) < g(n) i.e. f(n) = O(g(n))

but for this, second statement i.e. f(n) = Ω(g(n)) is incorrect.

Both statements are not true for all values of sin(n).