Question

Select the correct inequality for the asymptotic order of growth of the function n! – 2ⁿ?

1) n! – 2ⁿ = O(n!)
2) n! – 2ⁿ = O(2ⁿ)
3) n! – 2ⁿ = O(n²)
4) n! – 2ⁿ = O(n)

Answer 1

The correct Big O notation for the function n! − 2^n is O(n!), signifying that the factorial term dominates the asymptotic growth.

Step-by-step explanation:

You asked about the correct inequality for the asymptotic order of growth of the function n! − 2n. The correct inequality is: n! − 2n = O(n!). This is because the factorial function n! grows faster than any polynomial or exponential function such as 2n. As n increases, the 2n term becomes insignificant compared to the rapidly growing factorial term. Hence, the appropriate Big O notation reflects the dominant term, which in this case is n!.The correct inequality for the asymptotic order of growth of the function n! - 2ⁿ is n! - 2ⁿ = O(n!). This means that the function n! - 2ⁿ is upper bounded by the function n!. In other words, as n approaches infinity, the growth rate of n! - 2ⁿ is dominated by the growth rate of n!.