Question

Which of the following functions are O(x²)? Briefly explain the reasoning of your answer.

a. (x − 1)³
b. 2³ − 2x
c. 2^x + 1
d. 17x + 11

Answer 1

To determine which of the given functions are O(x²), compare their growth rates to x² as x approaches infinity.

Step-by-step explanation:

To determine which of the given functions are O(x²), we need to compare the growth rates of each function to x² as x approaches infinity.

a. (x − 1)³: This function has a higher degree than x², so it grows faster. Therefore, it is not O(x²).

b. 2³ − 2x: This is a linear function, which grows slower than x². Hence, it is O(x²).

c. 2^x + 1: Exponential functions always grow faster than any polynomial function like x², so it is not O(x²).

d. 17x + 11: This is a linear function, which grows slower than x². Hence, it is O(x²).

So, the functions b and d are O(x²).