Algorithms: Design and Analysis, Part 1 - Exam Question 9

Problem:

Let f and g be two increasing functions, defined on the natural numbers, with $f(1)$ , $g(1) \ge 1$. Assume that $f(n)=O(g(n))$ . Is $2^{f(n)}=O(2^{g(n)})$? (Multiple answers may be correct, check all that apply.)

Solution:

This is simply an assignment problem I have done before, it can be found in the link below.

Algorithms: Design and Analysis, Part 1 - Problem Set 1 - Question 3

The key idea is just going back to the definition and argue the necessary condition for $2^{f(n)}=O(2^{g(n)})$ is $f(n) \le g(n)$.