First, let's examine the given table of values for [tex]\( f(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-8 & -16 \\
\hline
-6 & -8 \\
\hline
-4 & 0 \\
\hline
-2 & 8 \\
\hline
0 & 16 \\
\hline
2 & 32 \\
\hline
4 & 64 \\
\hline
6 & 128 \\
\hline
\end{array}
\][/tex]
We need to determine the interval over which the function [tex]\( f(x) \)[/tex] is negative.
1. Observe the values in the second column which represent [tex]\( f(x) \)[/tex].
2. We need to identify the [tex]\( x \)[/tex]-values for which [tex]\( f(x) \)[/tex] is less than 0.
By looking at the table:
- When [tex]\( x = -8 \)[/tex], [tex]\( f(x) = -16 \)[/tex] (negative)
- When [tex]\( x = -6 \)[/tex], [tex]\( f(x) = -8 \)[/tex] (negative)
- When [tex]\( x = -4 \)[/tex], [tex]\( f(x) = 0 \)[/tex] (not negative)
- When [tex]\( x = -2 \)[/tex] and beyond (i.e., [tex]\( x = -2, 0, 2, 4, 6 \)[/tex]), [tex]\( f(x) \)[/tex] is positive.
Therefore, the function [tex]\( f(x) \)[/tex] is negative in the interval from [tex]\( x = -8 \)[/tex] to [tex]\( x = -6 \)[/tex].
Given each of the choices, the correct one must include the interval [tex]\( (-8, -6) \)[/tex]:
A) [tex]\((-8, -2)\)[/tex]: This is incorrect because [tex]\( f(x) \)[/tex] is positive for [tex]\( x = -2 \)[/tex].
B) [tex]\((-8, 0)\)[/tex]: This is incorrect because [tex]\( f(x) \)[/tex] becomes non-negative at [tex]\( x = -4 \)[/tex] and positive at [tex]\( x = 0 \)[/tex].
C) [tex]\((-\infty, -6)\)[/tex]: This is incorrect because it suggests that [tex]\( f(x) \)[/tex] is negative from negative infinity to [tex]\( -6 \)[/tex], which isn't supported by the given data.
D) [tex]\((-\infty, -4)\)[/tex]: This is incorrect because [tex]\( f(x) \)[/tex] is not negative as [tex]\( x \)[/tex] approaches [tex]\( -4 \)[/tex].
Therefore, the correct interval where [tex]\( f(x) \)[/tex] is negative is:
[tex]\[
(-8, -6)
\][/tex]
