For all of the problems below, when asked to give an example, you should give a function mapping positive integers to positive integers.

Find (with proof) a function f_1 such that f_1(2n) is O(f_1(n)). Find (with proof) a function f_2 such that f_2(2n) is not O(f_2(n)). Prove that if f(n) is O(g(n)), and g(n) is O(h(n)), then f(n) is O(h(n)). Give a proof or a counterexample: if f is not O(g), then g is O(f). Give a proof or a counterexample: if f is o(g), then f is O(g).