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

all real numbers less than or equal to 4

Step-by-step explanation:

To obtain the range we require the value of the y- coordinate of the vertex.

Given

f(x) = - (x + 5)(x + 1) ← equate f(x) to zero to find the zeros, that is

- (x + 5)(x + 1) = 0

Equate each factor to zero and solve for x

x + 5 = 0 ⇒ x = - 5

x + 1 = 0 ⇒ x = - 1

The vertex lies on the axis of symmetry which is at the midpoint of the zeros

= = = - 3

Substitute x = - 3 into f(x) for y- coordinate of vertex

= - (- 3 + 5)(- 3 + 1) = - (2)(- 2) = 4

The vertex = (- 3, 4 )

and - (x + 5)(x + 1) = - (x² + 6x + 5) = - x² - 6x - 5

Since the coefficient of the x² term is - 1 < 0

then the vertex is a maximum ∩

Hence range is [ 4, - ∞ )

That is all real numbers less than or equal to 4