To solve this problem, we need to find the value of [tex]\( x \)[/tex] such that [tex]\((f \circ g)(x) = -8\)[/tex]. The composite function [tex]\( (f \circ g)(x) \)[/tex] means [tex]\( f(g(x)) \)[/tex].
Given the values for [tex]\( f(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & -4 & -2 & 0 & 8 \\
\hline
f(x) & -8 & -2 & 4 & 32 \\
\hline
\end{array}
\][/tex]
And the values for [tex]\( g(x) \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 & 5 \\
\hline
g(x) & -1 & -2 & -2 & -4 & -8 \\
\hline
\end{array}
\][/tex]
We need to find [tex]\( x \)[/tex] such that [tex]\( f(g(x)) = -8 \)[/tex]. Let's examine each available [tex]\( x \)[/tex] to see if [tex]\( f(g(x)) = -8 \)[/tex].
1. For [tex]\( x = 1 \)[/tex]:
[tex]\[
g(1) = -1 \quad \text{(from the table of } g(x) \text{)}
\][/tex]
Since [tex]\( g(1) = -1 \)[/tex], we need to see if [tex]\( f(-1) = -8 \)[/tex]. However, [tex]\(-1\)[/tex] is not in the domain of [tex]\( f(x) \)[/tex] provided in the table.
2. For [tex]\( x = 2 \)[/tex]:
[tex]\[
g(2) = -2 \quad \text{(from the table of } g(x) \text{)}
\][/tex]
Since [tex]\( g(2) = -2 \)[/tex], we need to see if [tex]\( f(-2) = -8 \)[/tex]. From the values given:
[tex]\[
f(-2) = -2 \quad (\text{not } -8)
\][/tex]
3. For [tex]\( x = 3 \)[/tex]:
[tex]\[
g(3) = -2 \quad \text{(from the table of } g(x) \text{)}
\][/tex]
Again, if [tex]\( g(3) = -2 \)[/tex], we need to see if [tex]\( f(-2) = -8 \)[/tex]. From the values given:
[tex]\[
f(-2) = -2 \quad (\text{not } -8)
\][/tex]
4. For [tex]\( x = 4 \)[/tex]:
[tex]\[
g(4) = -4 \quad \text{(from the table of } g(x) \text{)}
\][/tex]
Since [tex]\( g(4) = -4 \)[/tex], we need to see if [tex]\( f(-4) = -8 \)[/tex]. From the values given:
[tex]\[
f(-4) = -8
\][/tex]
This is the desired result, therefore:
[tex]\[
f(g(4)) = -8.
\][/tex]
So, the value of [tex]\( x \)[/tex] when [tex]\((f \circ g)(x) = -8\)[/tex] is [tex]\( x = 4 \)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{4}
\][/tex]
