Given the information about the functions $f_1(n)$ and $f_2(n)$, we can analyze the time complexity of their sum $f_1(n) + f_2(n)$.

Definitions:

• $f_1(n) = \Omega(n^2)$ and $O(n^5)$ means that $f_1(n)$ grows at least as fast as $n^2$ and at most as fast as $n^5$.
• $f_2(n) = \Omega(n^3)$ and $O(n^6)$ means that $f_2(n)$ grows at least as fast as $n^3$ and at most as fast as $n^6$.

Sum of Functions:

When adding two functions, the time complexity is dominated by the term that grows the fastest (in Big-O) and the term that grows the slowest (in Big-Omega).

1. Big-O (Upper Bound):

   • $f_1(n)$ is $O(n^5)$.
   • $f_2(n)$ is $O(n^6)$.
   • The sum $f_1(n) + f_2(n)$ will be dominated by $f_2(n)$ because $n^6$ grows faster than $n^5$.
   • Therefore, $f_1(n) + f_2(n)$ is $O(n^6)$.

2. Big-Omega (Lower Bound):

   • $f_1(n)$ is $\Omega(n^2)$.
   • $f_2(n)$ is $\Omega(n^3)$.
   • The sum $f_1(n) + f_2(n)$ will be dominated by $f_2(n)$ at the lower bound because $n^3$ grows faster than $n^2$.
   • Therefore, $f_1(n) + f_2(n)$ is $\Omega(n^3)$.

Conclusion:

The correct answer is:

• $f_1(n) + f_2(n)$ is $O(n^6)$ and $\Omega(n^3)$.

Would you like more details on this analysis or have any questions?

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Here are 5 related questions you might find interesting:

1. How does the behavior of Big-O and Big-Omega differ when dealing with sums of functions?
2. What is the time complexity of the product of two functions, $f_1(n) \times f_2(n)$?
3. Can you provide examples of functions that exhibit similar Big-O and Big-Omega characteristics?
4. How does Big-Theta notation relate to Big-O and Big-Omega in this context?
5. What happens to the complexity if one of the functions is a constant, like $f_1(n) = O(1)$?

Tip: When analyzing the sum of functions, always consider the dominant terms at both the upper and lower bounds to determine the overall complexity.