To find [tex]\((f \circ g)(12)\)[/tex], we need to evaluate the inner function [tex]\(g\)[/tex] at 12 first, and then use the result as the input for the function [tex]\(f\)[/tex]. Here's the step-by-step process:
1. Evaluate [tex]\(g(12)\)[/tex]:
From the table for function [tex]\(g\)[/tex]:
[tex]\[
\begin{array}{c|rrrrrr}
x & -24 & -4 & 0 & 4 & 12 & 20 \\
\hline
g(x) & 19 & 4 & 1 & -2 & -8 & -14 \\
\end{array}
\][/tex]
When [tex]\(x = 12\)[/tex], [tex]\(g(12) = -8\)[/tex].
2. Evaluate [tex]\(f\)[/tex] at the result of [tex]\(g(12)\)[/tex]:
Now that we know [tex]\(g(12) = -8\)[/tex], we need to find [tex]\(f(-8)\)[/tex]. From the table for function [tex]\(f\)[/tex]:
[tex]\[
\begin{array}{c|rrrrrr}
x & -8 & -5 & -1 & 1 & 5 & 12 \\
\hline
f(x) & 16 & 10 & 2 & -2 & -10 & -24 \\
\end{array}
\][/tex]
When [tex]\(x = -8\)[/tex], [tex]\(f(-8) = 16\)[/tex].
So, the value of [tex]\((f \circ g)(12)\)[/tex] is [tex]\(f(g(12)) = 16\)[/tex].
The final answer is [tex]\((f \circ g)(12) = 16\)[/tex].
