Final answer:
The asymptotic bound for f(n)=7n+3n² is Θ(n²), for g(n)=n log n+8n it is Θ(n log n), and for h(n)=2 log n+5n+9 it is Θ(n).
Step-by-step explanation:
To determine the asymptotic bound of the functions f(n), g(n), and h(n), we evaluate the terms that grow the fastest as the value of n increases to infinity. This is often referred to as the dominant term and is used for simplifying the expression to its most significant component when n is very large (asymptotically).
For the function f(n) = 7n + 3n², the term 3n² grows faster than 7n as n approaches infinity, so the asymptotic bound is Θ(n²).
The function g(n) = n log n + 8n has n log n as its dominant term, since n log n grows faster than linear terms for large values of n. Thus, the asymptotic bound for g(n) is Θ(n log n).
Finally, the function h(n) = 2 log n + 5n + 9 has 5n as the dominant term, because it grows faster than both log n and the constant term. Therefore, the asymptotic bound for h(n) is Θ(n).