Question

The graph of the function f(x) = (x + 2)(x + 6) is shown below.

Which statement about the function is true?

The function is positive for all real values of x where
x > –4.
The function is negative for all real values of x where
–6 < x < –2.
The function is positive for all real values of x where
x < –6 or x > –3.
The function is negative for all real values of x where
x < –2.

Answer 1

The function is negative for all real values of x where

–6 < x < –2

Explanation:

the function f(x) = (x + 2)(x + 6)

LEts find the x intercept of the given function

0 = (x+2)(x+6)

x+2=0 so x=-2

x+6 = , so x=-6

x intercepts are -6 and -2

Lets draw a number line and use x intercepts and make 3 intervals

- infinity to -6 , -6 to -2 and -2 to infinity

LEts pick a number from each interval and check with f(x)

x<-6, pick -7

f(x) = (x + 2)(x + 6)

f(-7)=(-7 + 2)(-7 + 6) =5 , which is positive

–6 < x < –2, pick 4

f(-4)=(-4 + 2)(-4 + 6) =-4 , which is negative

x>-2, pick 0

f(0)=(-0+ 2)(0 + 6) =12 , which is positive

The function is negative for all real values of x where

–6 < x < –2

Answer 2

Explanation:

f(x) = (x + 2)(x +6)

1) The function is positive for all real values of x where x > –4 :

COUNTER-EXAMPLE : x = - 3 you have -3>-4 but (-3+2)(-3+6) = -1 ×3 =-3 no positive .

2) The function is positive for all real values of x where

x < –6 or x > –3.

COUNTER-EXAMPLE : x = - 2.5 you have -2.5>-3 but (-2.5+2)(-2.5+6) = -0.5 ×3.5 =-1.75 no positive .

same method for the statement : "The function is negative for all real values of x where

x < –2."

conclusion : statement about the function is true: "The function is negative for all real values of x where

–6 < x < –2."

.