Question

Give a big- O estimate for the function n²ⁿ+nⁿ² . For the function g(x) in your estimate that f(x) is O(g(x)), nse a simple function g of the smallest order.

(a) O(nⁿ²)
(b) O(2ⁿ )
(c) O(n²ⁿ )
(d) O(n²)
(c) O(nⁿ )

Answer 1

For the function n²ⁿ + nⁿ², the term n²ⁿ grows faster than nⁿ², and thus determines the function's growth rate as n becomes large. Consequently, the correct Big-O estimate for the function is O(n²ⁿ).

Step-by-step explanation:

The function we are analyzing for a Big-O estimate is n²ⁿ + nⁿ². When considering Big-O notation, we are interested in the upper bound of the growth of the function as n becomes very large. In the provided function, both terms n²ⁿ and nⁿ² contribute to the growth, but as n grows, one term will dominate the other in terms of the rate of growth.

Here, n²ⁿ is exponentially larger than nⁿ² because the base of the exponent in the first term is n², which grows faster than n to the power of another n as n becomes large. Therefore, when providing a Big-O estimate for the function n²ⁿ + nⁿ², we choose the term that grows the fastest as n increases, which in this case is n²ⁿ.

Therefore, the correct Big-O estimate for the function is O(n²ⁿ), which suggests that as n approaches infinity, the function's growth is limited above by some constant multiple of n²ⁿ. Hence, the answer is option (c) O(n²ⁿ).