Question

What is the domain and range of the quadratic function f(x) = -6(x + 4)² - 8?

a) Domain: All real numbers; Range: All real numbers
b) Domain: All real numbers; Range: y ≤ -8
c) Domain: x ≥ -4; Range: y ≥ -8
d) Domain: All real numbers; Range: y ≥ -8

Answer 1

The domain of the quadratic function f(x) = -6(x + 4)² - 8 is all real numbers and the range is y ≤ -8, because the parabola opens downwards and has a maximum value at its vertex.

Step-by-step explanation:

The function in question is f(x) = -6(x + 4)² - 8. This is a quadratic function in vertex form, where the vertex of the parabola is at (-4, -8) and the parabola opens downwards because the coefficient of the squared term is negative.

The domain of any quadratic function is always all real numbers, because there is no x-value for which the function is undefined. The range, however, depends on the direction in which the parabola opens. Since the coefficient of the squared term is negative, the function has a maximum value at the vertex. Therefore, the range of f(x) is all real numbers that are less than or equal to -8, or y ≤ -8.

So, the correct answer is: Domain: All real numbers; Range: y ≤ -8.