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."
