Let's analyze the function [tex]\( f(x) = (x + 2)(x + 6) \)[/tex] to determine its behavior at different intervals.
1. Identify the roots of the function:
The function [tex]\( f(x) = (x + 2)(x + 6) \)[/tex] is a quadratic expression which can be factored to find the roots.
[tex]\[ f(x) = 0 \][/tex]
[tex]\[ (x + 2)(x + 6) = 0 \][/tex]
The roots are:
[tex]\[ x = -2 \][/tex]
[tex]\[ x = -6 \][/tex]
2. Test the intervals defined by the roots:
The roots divide the real line into three intervals:
- [tex]\( x < -6 \)[/tex]
- [tex]\( -6 < x < -2 \)[/tex]
- [tex]\( x > -2 \)[/tex]
3. Determine the sign of the function in each interval:
- For [tex]\( x < -6 \)[/tex]:
Pick a test point, say [tex]\( x = -7 \)[/tex].
[tex]\[ f(-7) = (-7 + 2)(-7 + 6) = -5 \cdot -1 = 5 \][/tex]
The function is positive in this interval.
- For [tex]\( -6 < x < -2 \)[/tex]:
Pick a test point, say [tex]\( x = -5 \)[/tex].
[tex]\[ f(-5) = (-5 + 2)(-5 + 6) = -3 \cdot 1 = -3 \][/tex]
The function is negative in this interval.
- For [tex]\( x > -2 \)[/tex]:
Pick a test point, say [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = (0 + 2)(0 + 6) = 2 \cdot 6 = 12 \][/tex]
The function is positive in this interval.
4. Summarize the findings:
- The function [tex]\( f(x) = (x + 2)(x + 6) \)[/tex] is positive for [tex]\( x < -6 \)[/tex] and [tex]\( x > -2 \)[/tex].
- The function is negative for [tex]\( -6 < x < -2 \)[/tex].
Given these observations, the correct statement is:
The function is negative for all real values of [tex]\( x \)[/tex] where [tex]\( -6 < x < -2 \)[/tex].
