The graph of the function $f(x)=-(x+3)(x-1)$ 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 $-3 \leq x$
B. The domain is all real numbers such that $-3 \leq x \leq 1$, 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.
Answer (1)
To solve this problem, let's first understand what the function f ( x ) = − ( x + 3 ) ( x − 1 ) represents.
This is a quadratic function in factored form: − ( x + 3 ) ( x − 1 ) . The standard form of a quadratic function is a x 2 + b x + c . Here, you can expand − ( x + 3 ) ( x − 1 ) to find this form.
Expanding the function:
− ( x + 3 ) ( x − 1 ) = − ( x 2 − x + 3 x − 3 ) = − ( x 2 + 2 x − 3 ) = − x 2 − 2 x + 3
This is a downward-opening parabola because the leading coefficient ( a = − 1 ) is negative.
1. Domain:
For quadratic functions, unless specified by the problem, the domain is all real numbers. This is because there are no restrictions on the values that x can take.
2. Range:
Since this parabola opens downward, it has a maximum point at its vertex.
To find the vertex, use the formula for the vertex of a parabola in the form of a x 2 + b x + c , which is:
x = 2 a − b
Substitute a = − 1 and b = − 2 into the formula:
x = 2 ( − 1 ) − ( − 2 ) = − 2 2 = − 1
Substitute x = − 1 back into the function f ( x ) to find the maximum value (the y -coordinate of the vertex):
f ( − 1 ) = − (( − 1 ) + 3 ) (( − 1 ) − 1 ) = − ( 2 ) ( − 2 ) = 4
Thus, the range is all real numbers less than or equal to 4.
In conclusion:
Domain: All real numbers
Range: All real numbers less than or equal to 4.
The correct multiple-choice option that matches this is C : "The domain is all real numbers, and the range is all real numbers less than or equal to 4."
