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

Which statement about the function is true?

A. The function is positive for all real values of [tex]x[/tex] where [tex]x \ \textgreater \ -4[/tex].
B. The function is negative for all real values of [tex]x[/tex] where [tex]-6 \ \textless \ x \ \textless \ -2[/tex].
C. The function is positive for all real values of [tex]x[/tex] where [tex]x \ \textless \ -6[/tex] or [tex]x \ \textgreater \ -3[/tex].
D. The function is negative for all real values of [tex]x[/tex] where [tex]x \ \textless \ -2[/tex].


Sagot :

To solve this problem, let's carefully analyze the function \( f(x) = (x + 2)(x + 6) \).

### Step 1: Find the Roots of the Function

The roots of \( f(x) \) are the values of \( x \) that make \( f(x) \) equal to zero. We set the function equal to zero and solve for \( x \):

[tex]\[ (x + 2)(x + 6) = 0 \][/tex]

This equation will be zero if either \( x + 2 = 0 \) or \( x + 6 = 0 \). Thus, the roots are:

[tex]\[ x = -2 \][/tex]
[tex]\[ x = -6 \][/tex]

These roots divide the number line into three intervals:
1. \( (-\infty, -6) \)
2. \( (-6, -2) \)
3. \( (-2, \infty) \)

### Step 2: Determine the Sign of \( f(x) \) in Each Interval

#### Interval: \( (-\infty, -6) \)

Choose a test point in this interval. For example, \( x = -7 \):

[tex]\[ f(-7) = (-7 + 2)(-7 + 6) = (-5)(-1) = 5 \][/tex]

Since the value is positive, \( f(x) \) is positive in this interval.

#### Interval: \( (-6, -2) \)

Choose a test point in this interval. For example, \( x = -4 \):

[tex]\[ f(-4) = (-4 + 2)(-4 + 6) = (-2)(2) = -4 \][/tex]

Since the value is negative, \( f(x) \) is negative in this interval.

#### Interval: \( (-2, \infty) \)

Choose a test point in this interval. For example, \( x = 0 \):

[tex]\[ f(0) = (0 + 2)(0 + 6) = (2)(6) = 12 \][/tex]

Since the value is positive, \( f(x) \) is positive in this interval.

### Step 3: Summarize the Findings

Based on the above intervals and the corresponding signs of \( f(x) \), we have:
- The function \( f(x) \) is positive in the intervals \( (-\infty, -6) \) and \( (-2, \infty) \).
- The function \( f(x) \) is negative in the interval \( (-6, -2) \).

### Conclusion

By analyzing all the intervals, we can conclude that the correct statement is:

The function is negative for all real values of \( x \) where \( -6 < x < -2 \).

Therefore, the correct answer is:

The function is negative for all real values of [tex]\( x \)[/tex] where [tex]\( -6 < x < -2 \)[/tex].