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

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).

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