Question

Which of the following statements correctly expresses the relationship between f(n) = 2ⁿ and g(n) = 3ⁿ?

a. f(n) is O(g(n))
b. g(n) is O(f(n))
c. f(n) is Ω(g(n))
d. g(n) is Ω(f(n))

Answer 1

The correct relationship between the functions f(n) = 2^n and g(n) = 3^n is that f(n) is O(g(n)), which indicates that the growth rate of f(n) is bounded above by a constant multiple of g(n) as n increases.

Step-by-step explanation:

The relationship between f(n) = 2^n and g(n) = 3^n involves understanding how these functions grow relative to each other as n increases. To compare their growth rates, we use Big O, Big Omega, and Big Theta notations which are part of computational complexity theory in computer science. In this context, we say that f(n) is O(g(n)) if there exists a constant C and a value n0 such that f(n) ≤ C × g(n) for all n ≥ n0. Similarly, g(n) is Ω(f(n)) if there exists a constant C and a value n0 such that g(n) ≥ C × f(n) for all n ≥ n0.

In comparing the two functions, since 3^n grows faster than 2^n, we can deduce that f(n) = 2^n is O(g(n)) = 3^n which means that the growth rate of f(n) is bounded above by a constant multiple of g(n). Hence, the correct answer is option a. f(n) is O(g(n)).