Question

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

Answer 1

we have

$f(x)=-(x+3)(x-1)$

$f(x)=-(x^{2} -x+3x-3) \\ \\f(x)=-(x^{2} +2x-3)\\ \\f(x)=-x^{2}-2x+3$

Convert into vertex form

Group terms that contain the same variable, and move the constant to the opposite side of the equation

$f(x)-3=-(x^{2}+2x)$

Complete the square. Remember to balance the equation by adding the same constants to each side

$f(x)-3-1=-(x^{2}+2x+1)$

$f(x)-4=-(x^{2}+2x+1)$

Rewrite as perfect squares

$f(x)-4=-(x+1)^{2}$

$f(x)=-(x+1)^{2}+4$ -------> equation in vertex form

This is the equation of a vertical parabola open down

The vertex is a maximum---------> $(-1,4)$

therefore

the domain of the function is the interval---------> (-∞,∞)

Domain is all real numbers

The range of the function is the interval --------> (-∞,4]

Range is all real number less than or equal to $4$

Using a graphing tool

see the attached figure to better understand the problem

Answer 2

The domain is all real numbers.

The range is all real numbers less than or equal to 4.