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The graph of the function [tex]f(x)=-(x+3)(x-1)[/tex] is shown below.

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

A. The domain is all real numbers less than or equal to 4, and the range is all real numbers such that [tex]-3 \leq x \leq 1[/tex].
B. The domain is all real numbers such that [tex]-3 \leq x \leq 1[/tex], and the range is all real numbers less than or equal to 4.
C. The domain is all real numbers, and the range is all real numbers less than or equal to 4.
D. The domain is all real numbers less than or equal to 4, and the range is all real numbers.


Sagot :

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.