Question

the graph of the function $f(x) = (x +2)(x + 6)$ is shown below.
what is true about the domain and range of the function?
○ the domain is all real numbers, and the range is all real numbers greater than or equal to $-4$.
○ the domain is all real numbers greater than or equal to $-4$, and the range is all real numbers.
○ the domain is all real numbers such that $-6 \leq x \leq -2$, and the range is all real numbers greater than or equal to $-4$.
○ the domain is all real numbers greater than or equal to $-4$, and the range is all real numbers such that $-6 \leq x \leq -2$.

Explanation:

Step1: Identify function type

The function $f(x)=(x+2)(x+6)$ is a quadratic polynomial, which expands to $f(x)=x^2+8x+12$. Quadratic functions have no restrictions on input real numbers, so the domain is all real numbers.

Step2: Find vertex (minimum value)

For a quadratic function in the form $f(x)=ax^2+bx+c$, the x-coordinate of the vertex is $x=-\frac{b}{2a}$. Here, $a=1$, $b=8$, so:
$x=-\frac{8}{2\times1}=-4$
Substitute $x=-4$ into the function to find the minimum y-value:
$f(-4)=(-4+2)(-4+6)=(-2)(2)=-4$
Since $a=1>0$, the parabola opens upward, so the range is all real numbers greater than or equal to -4.

Step3: Match with options

This matches the first option.

Answer:

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