the graph of the function f(x)=-(x + 3)(x - 1) is shown below. what is true about the domain and range of…

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? the domain is all real numbers less than or equal to 4, and the range is all real numbers such that -3 ≤ x ≤ 1. the domain is all real numbers such that -3 ≤ x ≤ 1, and the range is all real numbers less than or equal to 4. the domain is all real numbers, and the range is all real numbers less than or equal to 4. the domain is all real numbers less than or equal to 4, and the range is all real numbers.

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? the domain is all real numbers less than or equal to 4, and the range is all real numbers such that -3 ≤ x ≤ 1. the domain is all real numbers such that -3 ≤ x ≤ 1, and the range is all real numbers less than or equal to 4. the domain is all real numbers, and the range is all real numbers less than or equal to 4. the domain is all real numbers less than or equal to 4, and the range is all real numbers.

Answer

Answer:

C. The domain is all real numbers, and the range is all real numbers less than or equal to 4.

Explanation:

Step1: Recall domain - definition

The domain of a polynomial function (like (f(x)=-(x + 3)(x - 1)=-x^{2}-2x + 3)) is all real numbers since we can substitute any real - valued (x) into the function.

Step2: Rewrite function in vertex form

[ \begin{align*} f(x)&=-(x + 3)(x - 1)\ &=- (x^{2}+2x - 3)\ &=-x^{2}-2x + 3\ &=-(x^{2}+2x+1 - 1)+3\ &=-((x + 1)^{2}-1)+3\ &=-(x + 1)^{2}+4 \end{align*} ]

Step3: Determine range

Since ((x + 1)^{2}\geq0) for all real (x), then (-(x + 1)^{2}\leq0) for all real (x). So, (f(x)=-(x + 1)^{2}+4\leq4). The range of the function is all real numbers (y) such that (y\leq4).