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

Explanation:

Step1: Recall domain - definition

The domain of a polynomial function is the set of all real numbers. The function $f(x)=-(x + 3)(x - 1)=-x^{2}-2x + 3$ is a quadratic polynomial, so the domain is all real numbers.

Step2: Find the vertex of the parabola

For a quadratic function in the form $y = ax^{2}+bx + c$ (here $a=-1$, $b=-2$, $c = 3$), the $x$-coordinate of the vertex is $x=-\frac{b}{2a}=-\frac{-2}{2\times(-1)}=-1$. Substitute $x = - 1$ into the function: $y=-(-1 + 3)(-1 - 1)=-2\times(-2)=4$. Since $a=-1<0$, the parabola opens downwards. So the maximum value of the function is $y = 4$, and the range is all real numbers less than or equal to 4.

Answer:

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