To determine the values of the function [tex]\( g \)[/tex] for [tex]\( g(-1) \)[/tex], [tex]\( g(-0.5) \)[/tex], and [tex]\( g(2) \)[/tex], we'll use the given piecewise definition of [tex]\( g \)[/tex] on the interval [tex]\((-2, 2]\)[/tex].
The function [tex]\( g(x) \)[/tex] is defined as:
[tex]\[
g(x) =
\begin{cases}
-1 & \text{ if } -2 < x \leq -1 \\
0 & \text{ if } -1 < x \leq 0 \\
1 & \text{ if } 0 < x \leq 1 \\
2 & \text{ if } 1 < x \leq 2
\end{cases}
\][/tex]
1. Finding [tex]\( g(-1) \)[/tex]:
- We need to see which interval [tex]\( -1 \)[/tex] falls into.
- According to the piecewise definition, for the interval [tex]\( -2 < x \leq -1 \)[/tex], the value of [tex]\( g(x) \)[/tex] is [tex]\(-1\)[/tex].
- Since [tex]\(-1\)[/tex] is included in [tex]\( -2 < x \leq -1 \)[/tex], thus:
[tex]\[
g(-1) = -1
\][/tex]
2. Finding [tex]\( g(-0.5) \)[/tex]:
- We need to see which interval [tex]\( -0.5 \)[/tex] falls into.
- According to the piecewise definition, for the interval [tex]\( -1 < x \leq 0 \)[/tex], the value of [tex]\( g(x) \)[/tex] is [tex]\(0\)[/tex].
- Since [tex]\(-0.5\)[/tex] is in the interval [tex]\( -1 < x \leq 0 \)[/tex], thus:
[tex]\[
g(-0.5) = 0
\][/tex]
3. Finding [tex]\( g(2) \)[/tex]:
- We need to see which interval [tex]\( 2 \)[/tex] falls into.
- According to the piecewise definition, for the interval [tex]\( 1 < x \leq 2 \)[/tex], the value of [tex]\( g(x) \)[/tex] is [tex]\(2\)[/tex].
- Since [tex]\(2\)[/tex] is included in [tex]\(1 < x \leq 2\)[/tex], thus:
[tex]\[
g(2) = 2
\][/tex]
In summary, the values of the function [tex]\( g \)[/tex] at the specified points are:
[tex]\[
g(-1) = -1, \quad g(-0.5) = 0, \quad g(2) = 2
\][/tex]
