Question

The function f(x) = −(x + 5)(x + 1) is shown.What is the range of the function?

all real numbers less than or equal to 4
all real numbers less than or equal to −3
all real numbers greater than or equal to 4
all real numbers greater than or equal to −3

Answer 1

The function f(x) = −(x + 5)(x + 1) is shown.What is the range of the function?

all real numbers less than or equal to 4

Answer 2

we have

`f(x)=-(x+5)(x+1)`

we know that

The equation of a vertical parabola into vertex form is equal to

`y=a(x-h)^(2)+k`

where

(h,k) is the vertex of the parabola

if
`a>0` ----> the parabola open upward (vertex is a minimum)

if
`a<0` ----> the parabola open downward (vertex is a maximum)

In this problem convert the quadratic equation into vertex form

so

`f(x)=-(x+5)(x+1)\\f(x)=-(x^(2) +x+5x+5)\\f(x)=-( x^(2) +6x+5)`

`f(x)=-x^(2)-6x-5`

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

`f(x)+5=-(x^(2)+6x)`

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

`f(x)+5-9=-(x^(2)+6x+9)`

`f(x)-4=-(x^(2)+6x+9)`

Rewrite as perfect squares

`f(x)-4=-(x+3)^(2)`

`f(x)=-(x+3)^(2)+4`

This is a vertical parabola open down (vertex is a maximum)

the vertex is the point
`(-3,4)`

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

`y\leq 4`

All real numbers less than or equal to
`4`

therefore

the answer is the option

All real numbers less than or equal to
`4`