To find the values of [tex]\( g(-1) \)[/tex], [tex]\( g(1) \)[/tex], and [tex]\( g(3) \)[/tex], we can refer directly to the definition of the function [tex]\( g(x) \)[/tex] for different domains of [tex]\( x \)[/tex]. Let's go step by step:
1. Find [tex]\( g(-1) \)[/tex]:
- For [tex]\( g(-1) \)[/tex], we need to determine which part of the function [tex]\( g(x) \)[/tex] to use. Since [tex]\(-2 < -1 \leq 1\)[/tex], we use the second piece of the function: [tex]\( g(x) = (x - 1)^2 \)[/tex].
- Now, substitute [tex]\( x = -1 \)[/tex] into this piece:
[tex]\[
g(-1) = (-1 - 1)^2 = (-2)^2 = 4
\][/tex]
2. Find [tex]\( g(1) \)[/tex]:
- For [tex]\( g(1) \)[/tex], [tex]\(-2 < 1 \leq 1\)[/tex] places us in the same primary domain as before: [tex]\( g(x) = (x - 1)^2 \)[/tex].
- Now, substitute [tex]\( x = 1 \)[/tex]:
[tex]\[
g(1) = (1 - 1)^2 = 0^2 = 0
\][/tex]
3. Find [tex]\( g(3) \)[/tex]:
- For [tex]\( g(3) \)[/tex], since [tex]\( 3 > 1 \)[/tex], we need to use the third part of the function: [tex]\( g(x) = 3 \)[/tex].
- Substitute [tex]\( x = 3 \)[/tex]:
[tex]\[
g(3) = 3
\][/tex]
Summarizing our findings, we have:
[tex]\[
\begin{array}{l}
g(-1) = 4 \\
g(1) = 0 \\
g(3) = 3
\end{array}
\][/tex]
Thus, the values are:
[tex]\[
g(-1) = 4, \quad g(1) = 0, \quad g(3) = 3
\][/tex]
