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Select the correct answer.

Consider functions [tex][tex]$f$[/tex][/tex] and [tex][tex]$g$[/tex][/tex].

\begin{tabular}{|c|c|c|c|c|}
\hline
[tex][tex]$x$[/tex][/tex] & -4 & -2 & 0 & 8 \\
\hline
[tex][tex]$f(x)$[/tex][/tex] & -8 & -2 & 4 & 32 \\
\hline
\end{tabular}

What is the value of [tex][tex]$x$[/tex][/tex] when [tex][tex]$(f \circ g)(x)=-8$[/tex][/tex]?

A. -4
B. 0
C. 3
D. 4

Sagot :

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]