To find the value of [tex]\( g(3) \)[/tex], we need to determine which piece of the piecewise function to use for [tex]\( x = 3 \)[/tex].
The function [tex]\( g(x) \)[/tex] is defined as follows:
[tex]\[ g(x) =
\begin{cases}
x - 1 & \text{for } -2 \leq x < -1 \\
2x + 3 & \text{for } -1 \leq x < 3 \\
6 - x & \text{for } x \geq 3
\end{cases}
\][/tex]
We are interested in the value of [tex]\( g(x) \)[/tex] when [tex]\( x = 3 \)[/tex]. Observing the conditions given in the piecewise function, we see that the appropriate piece to use is [tex]\( 6 - x \)[/tex], since it applies when [tex]\( x \geq 3 \)[/tex].
Now we substitute [tex]\( x = 3 \)[/tex] into the piece [tex]\( 6 - x \)[/tex]:
[tex]\[ g(3) = 6 - 3 = 3 \][/tex]
Hence, the value of [tex]\( g(3) \)[/tex] is:
[tex]\[ \boxed{3} \][/tex]
