Which of the following is false on two positive functions f and g?A. if f = ⇥ (g), then g = ⌦ (f )B. if f = ⇥ (g), then g = ⇥ (f )C. if f = ⇥ (g), then g = O(f )D. none of them.

For each of the following statements, mark in the answer sheets whether the statement istrue or false.(a) Let f and g be positive functions. Then, f (n) + g(n) ∈ O(n) implies that f ∈ O(n)and g ∈ O(n).(b) There exist functions f, g ∈ Ω(n) such that f − g ∈ Ω(g) and 2g − f ∈ Ω(f ).

Prove the following assertions by using the definitions of the notations in-volved, or disprove them by giving a specific counterexample.a. If t (n) ∈ O(g(n)), then g(n) ∈ (t (n)).b. (αg(n)) = (g(n)), where α > 0.c. (g(n)) = O(g(n)) ∩ (g(n)).d. For any two nonnegative functions t (n) and g(n) defined on the set ofnonnegative integers, either t (n) ∈ O(g(n)), or t (n) ∈ (g(n)), or both

If f(n) = O(g(n)), which of the following statements is true?g(n) grows faster than f(n) as n increasesf(n) grows faster than g(n) as n increasesf(n) and g(n) grow at the same rate as n increasesNone of the above

The following statements are equivalent:1 f ∈ Ω(g )2 g ∈ O(f )Proof. → Exercise

When performing the composition of two functions f(x)𝑓(𝑥) and g(x)𝑔(𝑥), [f∘g](x)[𝑓∘𝑔](𝑥)    and    [g∘f](x)[𝑔∘𝑓](𝑥) will produce the same answer.Question 5Select one:TrueFalse