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

Answer:

Detailed explanation:

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

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

COUNTER-EXAMPLE: x = -3 → -3>-4 but   (-3+2)(-3+6) = -1 ×3 =-3, not positive.

2) The function is positive for all real x where

x < –6 or x > –3.

COUNTER-EXAMPLE: x = -2.5 → -2.5>-3 but   (-2.5+2)(-2.5+6) = -0.5 ×3.5 =-1.75, not positive.

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

x < –2."

conclusion: the assertion regarding the function is valid: "The function is negative for all real x where

–6 < x < –2."

.

Answer:

The function takes negative values for all real x where

–6 < x < –2

Detailed explanation:

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

Let's find the x intercepts of the function

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

x+2=0, resulting in x=-2

x+6 = 0 giving x=-6

The x intercepts are -6 and -2

Now, we will draw a number line using the x intercepts, creating 3 intervals

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

Next, select a number from each interval to evaluate with f(x)

For x<-6, choose -7

f(-7) = (x + 2)(x + 6)

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

–6 < x < –2, choose -4

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

For x>-2, choose 0

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

The function is negative for all real x where

–6 < x < –2