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

Explanation:

Step1: Recall domain definition

For a polynomial function like $f(x)=(x + 2)(x+6)=x^{2}+8x + 12$, the domain is all real - numbers since there are no restrictions on the value of $x$ for which the function is undefined.

Step2: Find the vertex of the parabola

The function $y=x^{2}+8x + 12$ is a quadratic function in the form $y=ax^{2}+bx + c$ with $a = 1$, $b = 8$, $c = 12$. The $x$ - coordinate of the vertex is $x=-\frac{b}{2a}=-\frac{8}{2\times1}=-4$. Substitute $x=-4$ into the function: $y=(-4)^{2}+8\times(-4)+12=16-32 + 12=-4$. Since $a = 1>0$, the parabola opens upward, and the minimum value of the function is $y=-4$. So the range is all real numbers greater than or equal to $-4$.

Answer:

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