Final answer:
The domain and range of the function f(x) = -4√(3x - 12) are x ≥ 4 and f(x) ≤ 0, respectively.
Step-by-step explanation:
The question asks about the domain and range of the real-valued function f(x) = -4√(3x - 12). To find the domain, we need to determine the values of x for which the function is defined. The function includes a square root, so the expression inside the square root must be non-negative. Therefore, we need to solve the inequality 3x - 12 ≥ 0, which simplifies to x ≥ 4. Hence, the domain of the function is x ≥ 4.
The range of f(x) can be determined by examining the output values of the function knowing that the square root function outputs only non-negative values and the multiplication by -4 makes these values non-positive. Therefore, the range of f(x) is f(x) ≤ 0. Combining this information, we conclude that the correct answer is:
Domain: x ≥ 4; Range: f(x) ≤ 0