To determine the value of the function [tex]\( g(x) \)[/tex] when [tex]\( x = -1 \)[/tex]:
1. Start by substituting [tex]\( x = -1 \)[/tex] into the function [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = x^3 + 6x^2 + 12x + 8
\][/tex]
2. Now substitute [tex]\( x = -1 \)[/tex] into the function:
[tex]\[
g(-1) = (-1)^3 + 6(-1)^2 + 12(-1) + 8
\][/tex]
3. Next, calculate each term individually:
[tex]\[
(-1)^3 = -1
\][/tex]
[tex]\[
6(-1)^2 = 6 \times 1 = 6
\][/tex]
[tex]\[
12(-1) = -12
\][/tex]
[tex]\[
8 \text{ (constant term)}
\][/tex]
4. Add these results together:
[tex]\[
g(-1) = -1 + 6 - 12 + 8
\][/tex]
5. Perform the arithmetic step-by-step:
[tex]\[
\begin{align*}
-1 + 6 & = 5 \\
5 - 12 & = -7 \\
-7 + 8 & = 1
\end{align*}
\][/tex]
Therefore, the value of the function [tex]\( g(x) \)[/tex] when [tex]\( x = -1 \)[/tex] is [tex]\( g(-1) = 1 \)[/tex].
The correct answer is:
[tex]\[ g(-1) = 1 \][/tex]
