To solve this problem, we need to evaluate the piecewise function [tex]\( g(x) \)[/tex] at the given values of [tex]\( x \)[/tex]. The function [tex]\( g(x) \)[/tex] is defined as follows:
[tex]\[
g(x)=\left\{\begin{array}{ll}
x+4, & -5 \leq x \leq-1 \\
2-x, & -1
Let's evaluate [tex]\( g(x) \)[/tex] at the specified values step-by-step.
1. Evaluate [tex]\( g(-4) \)[/tex]:
- Since [tex]\( -4 \)[/tex] falls in the interval [tex]\( -5 \leq x \leq -1 \)[/tex], we use the first part of the piecewise function:
[tex]\[
g(-4) = -4 + 4 = 0
\][/tex]
2. Evaluate [tex]\( g(-2) \)[/tex]:
- Since [tex]\( -2 \)[/tex] falls in the interval [tex]\( -5 \leq x \leq -1 \)[/tex], we use the first part of the piecewise function again:
[tex]\[
g(-2) = -2 + 4 = 2
\][/tex]
3. Evaluate [tex]\( g(0) \)[/tex]:
- Since [tex]\( 0 \)[/tex] falls in the interval [tex]\( -1 < x \leq 5 \)[/tex], we use the second part of the piecewise function:
[tex]\[
g(0) = 2 - 0 = 2
\][/tex]
4. Evaluate [tex]\( g(3) \)[/tex]:
- Since [tex]\( 3 \)[/tex] falls in the interval [tex]\( -1 < x \leq 5 \)[/tex], we use the second part of the piecewise function:
[tex]\[
g(3) = 2 - 3 = -1
\][/tex]
5. Evaluate [tex]\( g(4) \)[/tex]:
- Since [tex]\( 4 \)[/tex] falls in the interval [tex]\( -1 < x \leq 5 \)[/tex], we use the second part of the piecewise function:
[tex]\[
g(4) = 2 - 4 = -2
\][/tex]
Thus, the evaluations of [tex]\( g(x) \)[/tex] at the specified points are as follows:
[tex]\[
\begin{array}{l}
g(-4) = 0 \\
g(-2) = 2 \\
g(0) = 2 \\
g(3) = -1 \\
g(4) = -2
\end{array}
\][/tex]
So, the final results are:
[tex]\[
( g(-4), g(-2), g(0), g(3), g(4) ) = ( 0, 2, 2, -1, -2 )
\][/tex]
