Certainly! Let's determine the values of [tex]\( g(x) \)[/tex] over the interval from [tex]\( -5 \)[/tex] to [tex]\( 5 \)[/tex] based on the piecewise function given.
The function [tex]\( g(x) \)[/tex] is defined as follows:
[tex]\[
g(x)=\begin{cases}
x + 4 & \text{for } -5 \leq x \leq -1 \\
2 - x & \text{for } -1 < x \leq 5
\end{cases}
\][/tex]
We will evaluate [tex]\( g(x) \)[/tex] at each integer value of [tex]\( x \)[/tex] within the interval [tex]\([-5, 5]\)[/tex].
### For [tex]\( x \)[/tex] in the range [tex]\([-5, -1]\)[/tex]:
- When [tex]\( x = -5 \)[/tex]:
[tex]\[
g(-5) = -5 + 4 = -1
\][/tex]
- When [tex]\( x = -4 \)[/tex]:
[tex]\[
g(-4) = -4 + 4 = 0
\][/tex]
- When [tex]\( x = -3 \)[/tex]:
[tex]\[
g(-3) = -3 + 4 = 1
\][/tex]
- When [tex]\( x = -2 \)[/tex]:
[tex]\[
g(-2) = -2 + 4 = 2
\][/tex]
- When [tex]\( x = -1 \)[/tex]:
[tex]\[
g(-1) = -1 + 4 = 3
\][/tex]
### For [tex]\( x \)[/tex] in the range [tex]\((-1, 5]\)[/tex]:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
g(0) = 2 - 0 = 2
\][/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
g(1) = 2 - 1 = 1
\][/tex]
- When [tex]\( x = 2 \)[/tex]:
[tex]\[
g(2) = 2 - 2 = 0
\][/tex]
- When [tex]\( x = 3 \)[/tex]:
[tex]\[
g(3) = 2 - 3 = -1
\][/tex]
- When [tex]\( x = 4 \)[/tex]:
[tex]\[
g(4) = 2 - 4 = -2
\][/tex]
- When [tex]\( x = 5 \)[/tex]:
[tex]\[
g(5) = 2 - 5 = -3
\][/tex]
Now we have calculated the values for [tex]\( g(x) \)[/tex] at each integer within the interval [tex]\([-5, 5]\)[/tex]. The list of evaluated [tex]\( x \)[/tex] values and their corresponding [tex]\( g(x) \)[/tex] values are as follows:
- [tex]\( x \)[/tex] values: [tex]\([-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5]\)[/tex]
- [tex]\( g(x) \)[/tex] values: [tex]\([-1, 0, 1, 2, 3, 2, 1, 0, -1, -2, -3]\)[/tex]
This provides a clear and comprehensive evaluation of the piecewise function [tex]\( g(x) \)[/tex] over the specified domain.
