Final answer:
The domain of the function f(x) = √(-x² - 8x - 12) is found by determining where the expression under the square root is non-negative. The correct option is D .
Step-by-step explanation:
To find the domain of the function f(x) = √(-x² - 8x - 12), we need to consider the values of x for which the expression inside the square root is non-negative, since the square root of a negative number is not defined in the real number system.
Let's solve the inequality -x² - 8x - 12 ≥ 0. We will start by finding the roots of the corresponding quadratic equation -x² - 8x - 12 = 0 using the quadratic formula, where a = -1, b = -8, and c = -12.
Substituting these values into the quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), we get:
x = (8 ± √((-8)² - 4(-1)(-12))) / (2(-1))
x = (8 ± √(64 - 48)) / (-2)
x = (8 ± √(16)) / (-2)
x = (8 ± 4) / (-2)
So, we have the roots x = (-8 - 4) / (-2) = 6 and x = (-8 + 4) / (-2) = -2. Since the coefficient of x² is negative, our parabola opens downward, meaning the expression is non-negative between the roots.
Therefore, the domain of f(x) is the interval (-6, -2).
However, this does not match any of the given options. Thus, the student may have provided the incorrect function or the options may contain a mistake. If the student is certain about the provided options, it would be helpful to review the original problem or seek clarification.