Question

Assume again two (positive) nondecreasing functions fff and ggg such that f(n)=O(g(n))f(n)=O(g(n))f(n)=O(g(n)). Is 2f(n)=O(2g(n))2^{f(n)}=O(2^{g(n)})2f(n)=O(2g(n)) ? (Multiple answers may be correct, you should check all of those that apply.)

(1)Always

(2)Yes if f(n)≤g(n)f(n) \le g(n)f(n)≤g(n) for all sufficiently large nnn

(3)Never

(4)Sometimes yes, sometimes no (depending on fff and ggg)

Answer 1

Hello.

Here is your answer

Option 3 is the correct answer

I hope you are well

Thanks for the questioning

Answer 2

Assume again two (positive) nondecreasing functions fff and ggg such that f(n)=O(g(n))f(n)=O(g(n))f(n)=O(g(n)). Is 2f(n)=O(2g(n))2^{f(n)}=O(2^{g(n)})2f(n)=O(2g(n)) ? (Multiple answers may be correct, you should check all of those that apply.)

(1)Always

(2)Yes if f(n)≤g(n)f(n) \le g(n)f(n)≤g(n) for all sufficiently large nnn

(3)Never✔✔