Let's analyze the function [tex]\( f(x) = (x+2)(x+6) \)[/tex] step-by-step to determine its domain and range.
### Step 1: Identify the Domain
The function [tex]\( f(x) = (x+2)(x+6) \)[/tex] is a quadratic function. Quadratic functions are defined for all real values of [tex]\( x \)[/tex]. Hence, the domain of the function is:
- [tex]\( \text{Domain: All real numbers} \)[/tex]
### Step 2: Identify the Range
Next, we need to determine the range of the function. Since this is a quadratic function that opens upwards (the coefficient of the [tex]\( x^2 \)[/tex] term is positive when expanded), it has a minimum value at its vertex.
To find the vertex:
1. Identify the roots of the function: The roots of [tex]\( f(x) = (x+2)(x+6) \)[/tex] are [tex]\( x = -2 \)[/tex] and [tex]\( x = -6 \)[/tex].
2. The x-coordinate of the vertex is the midpoint of the roots:
[tex]\[
x = \frac{-2 + (-6)}{2} = \frac{-8}{2} = -4
\][/tex]
3. Substitute [tex]\( x = -4 \)[/tex] back into the function to find the y-coordinate of the vertex:
[tex]\[
f(-4) = (-4 + 2)(-4 + 6) = (-2)(2) = -4
\][/tex]
Thus, the vertex of the function is at [tex]\((-4, -4)\)[/tex], which is the minimum point.
Since the parabola opens upwards, the range of the function is all real numbers greater than or equal to [tex]\(-4\)[/tex].
- [tex]\( \text{Range: All real numbers greater than or equal to -4} \)[/tex]
### Conclusion
Putting it all together:
- The domain of [tex]\( f(x) = (x+2)(x+6) \)[/tex] is all real numbers.
- The range of [tex]\( f(x) \)[/tex] is all real numbers greater than or equal to -4.
Therefore, 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.
