To determine the values of the function [tex]\( g(x) \)[/tex] at specific points, we need to evaluate the function according to the defined piecewise conditions.
The piecewise function is defined as:
[tex]\[
g(x) = \left\{\begin{array}{ll}
6, & \text{if } -8 \leq x < -2 \\
0, & \text{if } -2 \leq x < 4 \\
-4, & \text{if } 4 \leq x < 10
\end{array}\right.
\][/tex]
Step-by-step:
1. Evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = -2 \)[/tex]:
- According to the piecewise function, if [tex]\(-2 \leq x < 4\)[/tex], then [tex]\( g(x) = 0 \)[/tex].
- Since [tex]\(-2\)[/tex] falls in the interval [tex]\([-2, 4)\)[/tex], we use the rule [tex]\( g(x) = 0 \)[/tex].
Therefore, [tex]\( g(-2) = 0 \)[/tex].
2. Evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = 4 \)[/tex]:
- According to the piecewise function, if [tex]\(4 \leq x < 10\)[/tex], then [tex]\( g(x) = -4 \)[/tex].
- Since [tex]\(4\)[/tex] falls in the interval [tex]\([4, 10)\)[/tex], we use the rule [tex]\( g(x) = -4 \)[/tex].
Therefore, [tex]\( g(4) = -4 \)[/tex].
So, the values of the function are:
[tex]\[
\begin{array}{l}
g(-2) = 0 \\
g(4) = -4
\end{array}
\][/tex]
