Question

Which of the following statements is true regarding the asymptotic growth rate of the sum of two functions, f(n) and g(n)?

a) If f(n) = O(h(n)) and g(n) = O(k(n)), then f(n) + g(n) = O(h(n) + k(n))
b) If f(n) = O(h(n)) and g(n) = Ω(k(n)), then f(n) + g(n) = Θ(h(n) + k(n))
c) If f(n) = Θ(h(n)) and g(n) = O(k(n)), then f(n) + g(n) = Ω(h(n) + k(n))
d) If f(n) = Ω(h(n)) and g(n) = Ω(k(n)), then f(n) + g(n) = O(h(n) + k(n))

Answer 1

The true statement regarding the asymptotic growth rate of function sums is that if f(n) is O(h(n)) and g(n) is O(k(n)), then their sum is also O(h(n) + k(n)). This pertains to algorithm analysis and the use of big O notation.

Step-by-step explanation:

The question relates to the asymptotic growth rate of functions, which is a core concept in algorithm analysis in computer science. To understand this concept, we need to familiarize ourselves with the big O notation (O), big omega notation (Ω), and big theta notation (Θ), which describe upper bounds, lower bounds, and tight bounds of an algorithm's running time or space requirements, respectively.

The correct statement regarding the asymptotic growth rate of the sum of two functions is (a): If f(n) = O(h(n)) and g(n) = O(k(n)), then f(n) + g(n) = O(h(n) + k(n)). This is because the big O notation specifies an upper bound, and when two functions are added together, their combined growth rate will be at most the sum of their individual growth rates. Therefore, the growth rate of the sum f(n) + g(n) will not be faster than the sum of the growth rates of the individual functions.