the graph of the function ( f(x)=(x + 2)(x + 6) ) is shown below. what is true about the domain and range of…

Explanation:

Step1: Analyze the domain of a quadratic function

For any quadratic function (y = ax^{2}+bx + c) (in this case (f(x)=(x + 2)(x + 6)=x^{2}+8x + 12)), the domain is all real numbers because there are no restrictions on the values of (x) for which the function is defined.

Step2: Find the vertex of the parabola (to determine the range)

The (x) - coordinate of the vertex of a parabola (y=ax^{2}+bx + c) is given by (x=-\frac{b}{2a}). Here (a = 1), (b = 8), so (x=-\frac{8}{2\times1}=-4). Substitute (x=-4) into the function (f(x)=(x + 2)(x + 6)): (f(-4)=(-4 + 2)(-4+6)=(-2)\times2=-4). Since (a = 1>0), the parabola opens upwards. So the minimum value of the function is (y=-4). The range of the function (y) (or (f(x))) is (y\geq - 4).

Answer:

The domain is all real numbers, and the range is all real numbers greater than or equal to (-4).