The following are the asymptotic growth rates as functions o

The following are the asymptotic growth rates as functions of the data size (n).

4n log(n) +2n, 2¹⁰, 2^log(n), 3n+1--log(n), 4n, 2ⁿ, n²+10n, n³, nlog(n²)

a) Write the Big-O order of each of these growth rates

b) Arrange these growth rates in ascending order (slowest to fastest)

Solution

I am expecting, Chegg\'s Text view got format of these expressions wrong
So, I am gonna assume a few things here..
5)   4^n instead of 4n
6)   2^n instead of 2n
7)   n^2+10n instead of n2+10n
8)   nlog(n^2) instead of nlog(n2)

1.   4nlog(n) + 2n    =>    O(nlog(n)), since nlog(n) is asymptotically larger than n and by removing constant terms.
2.   210            =>   O(1),        since this is independent of n.
3.   2log(n)           =>   O(log(n)),    by removing constant term.
4.   3n+1--log(n)   =>   O(n),       since n is asymptotically larger than log(n) and also by removing constant terms.
5.   4^n               =>   O(4^n)
6.   2^n               =>   O(2^n)
7.   n^2+10n           =>   O(n^2),       since n^2 is asymptotically larger than n and also by removing constant terms.
8.   n^3               =>   O(n^3)
9.   nlog(n^2)       =>   O(nlog(n)),   nlog(n^2) = 2nlog(n) and then by removing constant terms.

growth rates in ascending order:
Most of the time, you will have one of the following.
O(1) -> O(log(n)) -> O(n) -> O(nlog(n)) -> O(n^x) -> O(x^n) -> O(x^n) -> O(n^n)

2 -> 3 -> 4 -> 1, 9 -> 7 -> 8 -> 6 -> 5 is the order.