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

Answer:

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

Explanation:

Step1: Recall domain definition

For polynomial functions like $f(x)=(x + 2)(x+6)=x^{2}+8x + 12$, the domain is all real numbers since we can substitute any real - valued $x$ into the function without causing any undefined operations (like division by zero or taking the square root of a negative number in the basic form of the function).

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 $a = 1$, $b = 8$, and $c = 12$. The $x$ - coordinate of the vertex of a parabola $y=ax^{2}+bx + c$ is given by $x=-\frac{b}{2a}$. So, $x=-\frac{8}{2\times1}=-4$.

Step3: Find the $y$ - value of the vertex

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. So the range is all real numbers $y\geq - 4$.