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

Solution:

The domain of a function is defined as the set of all input values for the function f (x), whereas the range of a function is the set of all possible output values.

Given: f (x) = - (x + 5) (x + 1)

Let y = f (x) = - (x + 5) (x + 1)

To find the range of the function, we will substitute different values of x.

⇒ f (- 5 ) = - (-5 + 5) (-5 + 1) = - (0) (-4) = 0

⇒ f (- 4 ) = - (-4 + 5) (-4 + 1) = - (1) (-3) = 3

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

⇒ f (- 2 ) = -(-2 + 5) (-2 + 1) = - (3) (-1) = 3

⇒ f (- 1 ) = -(-1 + 5) (-1 + 1) = - (4) (0) = 0 

⇒ f ( 0 ) = -(0 + 5) (0 + 1) = - (5) (1) = - 5

⇒ f ( 1 ) = -(1 + 5) (1 + 1) = - (6) (2) = - 12

⇒ f ( 2 ) = -(2 + 5) (2 + 1) = - (7) (3) = - 21

Thus, the coordinates are (-5, 0), (-4, 3), (-3, 4), (-2, 3), (-1, 0), (0, -5), (1, -12), (2, -21)

Let's plot a graph using the above coordinates.

The graph of the function is shown below.

Thus, the range of the function f (x) = -(x + 5) (x + 1) is all real numbers less than or equal to 4.

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

Summary:

The range of the function f (x) = -(x + 5) (x + 1) is all real numbers less than or equal to 4.