Let's analyze the given function step-by-step to determine its domain and range.
The given function is [tex]\( f(x) = -(x + 3)(x - 1) \)[/tex].
### Domain:
1. Identify Restrictions:
The function [tex]\( f(x) = -(x + 3)(x - 1) \)[/tex] is a quadratic polynomial. Quadratic functions are defined for all real numbers because there are no values of [tex]\( x \)[/tex] that will make the function undefined.
2. Conclusion for Domain:
Therefore, the domain of [tex]\( f(x) \)[/tex] is all real numbers.
### Range:
1. Extract Information About the Shape of the Quadratic:
The function is in the form of a quadratic function, specifically [tex]\( f(x) = ax^2 + bx + c \)[/tex].
Given that the coefficient of the [tex]\( x^2 \)[/tex] term (which can be implied by expanding [tex]\( -(x+3)(x-1) \)[/tex]) is negative, it follows that the parabola opens downwards.
2. Find the Vertex:
The vertex of a downward-opening parabola is its maximum point. For a quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex], the x-coordinate of the vertex can be found by using the formula:
[tex]\[
x = \frac{-b}{2a}
\][/tex]
However, for this specific function, we can find the vertex by considering the symmetry of the roots [tex]\( x = -3 \)[/tex] and [tex]\( x = 1 \)[/tex].
The x-coordinate of the vertex is the average of the roots:
[tex]\[
x = \frac{-3 + 1}{2} = -1
\][/tex]
To find the y-coordinate of the vertex (maximum value of the function), substitute [tex]\( x = -1 \)[/tex] back into the function:
[tex]\[
f(-1) = -(-1 + 3)(-1 - 1) = -(2)(-2) = 4
\][/tex]
3. Conclusion for Range:
Since the vertex is the maximum point and the parabola opens downwards, the range of the function includes all real numbers less than or equal to the y-coordinate of the vertex.
Therefore, the range is all real numbers less than or equal to 4.
### Summary:
1. Domain: All real numbers.
2. Range: All real numbers less 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 less than or equal to 4.
