Please type out solution neatly

1.

For each of the following pairs of functions, either f(n) is O(g(n)), f(n) is (g(n)), or f(n) is (g(n)) best describes the relationship. Select one and explain.

a. f(n) = n^0.75; g(n) = n^0.5

b. f(n) = log n; g(n) = ln n

c. f(n) = nlog n; g(n) =n

d. f(n) = e^n; g(n) = 3^n

e. f(n) = 2^n; g(n) = 2^n-1

f. f(n) = 4^n; g(n) = n!

2

Let f1 and f2 be asymptotically positive non-decreasing functions. Prove or disprove each of the following conjectures. To disprove give a counter example.

a. If f1(n) = W(g(n)) and f2(n) = O(g(n)) then f1(n)= Q (f2(n) ).

b. If f1(n) = O(g1(n)) and f2(n) = O(g2(n)) then f1(n)+ f2(n)= O(g1(n) + g2(n) )