Final answer:
The Big-Oh complexities are O(n) for the for-loop, O(n) for the recursive function, and O(n^4 log(n)) for the function f(n). These estimates provide an upper bound on the runtime or the growth of the function for large inputs.
Step-by-step explanation:
The Big-Oh estimate provides an upper bound on the growth of a function for large values of n. It is a common way to express the worst-case complexity of an algorithm in computer science.
a. For-loop Complexity
The given Java for-loop runs
from i = 1 to n and contains a single statement that runs in constant time. The loop iterates (n-1) times, so the Big-Oh complexity is O(n).
b. Recursive Function Complexity
The recursive function 'f' will be executed 'n' times until the base case is reached. Each function call is a constant time operation, but because it calls itself 'n' times, the Big-Oh complexity is O(n).
c. Function Growth Rate
The function f(n) = (n+5)4 log(2n^4 + 5) can be simplified as f(n) = (n^4)log(n). The dominant term for large n is n^4 log(n), so the Big-Oh complexity is O(n^4 log(n)).