The graph of the function [tex]f(x) = (x + 2)(x + 6)[/tex] is shown below.

What is true about the domain and range of the function?

A. The domain is all real numbers, and the range is all real numbers greater than or equal to -4.
B. The domain is all real numbers greater than or equal to -4, and the range is all real numbers.
C. The domain is all real numbers such that [tex]-6 \leq x \leq -2[/tex], and the range is all real numbers greater than or equal to -4.
D. The domain is all real numbers greater than or equal to -4, and the range is all real numbers such that [tex]-6 \leq x \leq -2[/tex].



Answer :

To determine the domain and range of the function \( f(x) = (x + 2)(x + 6) \), we can follow these detailed steps:

1. Domain:
- The function \( f(x) = (x + 2)(x + 6) \) is a quadratic function. Quadratic functions are defined for all real numbers. Therefore, the domain of this function is all real numbers.

2. Range:
- To find the range, we first need to determine the vertex of the parabola described by the function \( f(x) \).
- We can rewrite the function in standard quadratic form \( f(x) = ax^2 + bx + c \). Expanding the given function, we have:
[tex]\[ f(x) = (x + 2)(x + 6) = x^2 + 8x + 12 \][/tex]
- The general form of a quadratic function is \( y = ax^2 + bx + c \), where \( a = 1 \), \( b = 8 \), and \( c = 12 \).

3. Vertex:
- The vertex of a parabola in the form \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \).
[tex]\[ x = -\frac{8}{2(1)} = -4 \][/tex]
- To find the \( y \)-coordinate of the vertex, substitute \( x = -4 \) back into the function:
[tex]\[ f(-4) = (-4 + 2)(-4 + 6) = (-2)(2) = -4 \][/tex]
- Therefore, the vertex of the parabola is \( (-4, -4) \).

4. Range:
- Since the coefficient of \( x^2 \) (which is \( a = 1 \)) is positive, the parabola opens upwards. This means that the vertex represents the minimum value of the function.
- Therefore, the range of the function is all real numbers greater than or equal to the \( y \)-coordinate of the vertex, which is \(-4\).

Combining these insights, we conclude:

- The domain of the function \( f(x) = (x + 2)(x + 6) \) is all real numbers.
- The range of the function is all real numbers greater than or equal to \(-4\).

Thus, the correct statement about the domain and range of the function is:
"The domain is all real numbers, and the range is all real numbers greater than or equal to -4."