Final Answer:
The expression n³ is O(g(n)) in the case of g(n) = n³ (Option C) since n³ is asymptotically equal to itself. For all other options (A, B, D), n³ is not O(g(n)) as they grow at a different rate compared to n³.
Step-by-step explanation:
Big-O notation (O) is used in the analysis of algorithms to describe the upper bound or worst-case scenario for the growth rate of a function concerning the input size (n). For the given options (Option C):
g(n) = n² (Option A): n³ grows faster than n², so n³ is not O(g(n)).
g(n) = n² + n³ (Option B): Here, the term n³ dominates, and n³ is the highest power, so n³ is not O(g(n)).
g(n) = n³ log₂ n (Option C): n³ is asymptotically equal to n³, so n³ is O(g(n)).
g(n) = n⁴/log₂ n (Option D): The logarithmic term log₂ n will eventually be surpassed by the polynomial term n⁴, making n³ not O(g(n)).
Understanding the growth rate of functions is crucial in algorithmic analysis to assess the efficiency and performance of algorithms as the input size increases. The notation O(g(n)) helps in categorizing the rate of growth, providing insights into the scalability and efficiency of algorithms under different scenarios.
In summary, for n³ to be O(g(n)), the function g(n) must not grow faster than n³. Among the given options, only g(n) = n³ log₂ n satisfies this condition, making the correct choice Option C.