Consider the following functions:

f(n) = 2^n g(n) = n! h(n) = n^logn

Which of the following statements about the asymptotic behavior of f(n), g(n), and h(n) is true?

(A) f(n) = O(g(n)); g(n) = O(h(n))

(B) f(n) = (g(n)); g(n) = O(h(n))

(C) g(n) = O(f(n)); h(n) = O(f(n))

(D) h(n) = O(f(n)); g(n) = (f(n))

**(A)** A

**(B)** B

**(C)** C

**(D)** D

**Answer:** **(D)** **Explanation:** According to order of growth: h(n) < f(n) < g(n) (g(n) is asymptotically greater than f(n) and f(n) is asymptotically greater than h(n) )

We can easily see above order by taking logs of the given 3 functions

lognlogn < n < log(n!) (logs of the given f(n), g(n) and h(n)).

Note that log(n!) = (nlogn)

Quiz of this Question