domain and range of the function:..?

Question

Find the domain and range of the function
f(x) = 2 x ^2 + 1, -1 ≤ x ≤ 2.

a. Domain [-2,2], Range [1,9]
b. Domain [-1,1], Range [2,9]
c. Domain [-1,2], Range [4,9]
d. Domain [-1,2], Range [2,∞]
e. Domain [-1,2], Range [1,9]

Answer 1

First, notice that the restrictions -1 <= x <= 2 tell you that the domain is the interval [-1,2]. The derivative is f'(x) = 4x, which vanishes at x = 0. We need to know the values of f(x) at x = 0 (the minimum), at x = -1, and at x = 2. The reason is that at the ends of a closed interval-domain, the function may attain a max or min without having derivative = 0.

We find f(-1) = 3, f(0) = 1, and f(2) = 9. Since the function is continuous throughout its domain, the range is [1,9]. Thus, the correct answer is e.

Answer 2

e

0 give you the lowest number, 1
2 gives you the highest number, 9