To determine the function's value when [tex]\(x = -1\)[/tex] for the function [tex]\( g(x) = x^3 + 6x^2 + 12x + 8 \)[/tex], we can follow these steps:
1. Substitute [tex]\( x = -1 \)[/tex] into the function:
[tex]\[ g(-1) = (-1)^3 + 6(-1)^2 + 12(-1) + 8 \][/tex]
2. Calculate each term separately:
- [tex]\((-1)^3 = -1\)[/tex]
- [tex]\(6(-1)^2 = 6 \cdot 1 = 6\)[/tex]
- [tex]\(12(-1) = -12\)[/tex]
- The constant [tex]\(8\)[/tex] remains [tex]\(8\)[/tex]
3. Combine these results:
[tex]\[ g(-1) = -1 + 6 - 12 + 8 \][/tex]
4. Perform the addition and subtraction step by step:
- First, combine [tex]\(-1\)[/tex] and [tex]\(6\)[/tex]:
[tex]\[ -1 + 6 = 5 \][/tex]
- Next, subtract [tex]\(12\)[/tex] from [tex]\(5\)[/tex]:
[tex]\[ 5 - 12 = -7 \][/tex]
- Finally, add [tex]\(8\)[/tex] to [tex]\(-7\)[/tex]:
[tex]\[ -7 + 8 = 1 \][/tex]
So, the value of the function [tex]\( g(x) \)[/tex] when [tex]\( x = -1 \)[/tex] is [tex]\( g(-1) = 1 \)[/tex].
Therefore, the correct answer is:
[tex]\[ g(-1) = 1 \][/tex]
