Analyze the time complexity of the following functions using Big-O Notation, and explain the concept of an intractable problem with an example.

c)

iii) Given the tight upper bound for the algorithm with the time upper bound functions. Use Big-O Notation for your answers:

1. $n^2 + 3n + 1$: The dominant term is $n^2$. $\boxed{O(n^2)}$
2. $2n + \log n$: The dominant term is $2n$. $\boxed{O(n)}$
3. $n + n^3 + 2^n$: The dominant term is $2^n$. $\boxed{O(2^n)}$

d) An intractable problem is a computational problem for which no efficient algorithm exists to solve it in a reasonable amount of time, especially as the input size grows. Typically, these problems have a time complexity that is exponential or factorial, meaning the time required to solve them becomes astronomically large for even moderately sized inputs.

Example: The Traveling Salesperson Problem (TSP). Given a list of cities and the distances between each pair of cities, the goal is to find the shortest possible route that visits each city exactly once and returns to the origin city. The number of possible routes grows factorially with the number of cities, making it intractable for a large number of cities.