To determine the value of [tex]\( g(3) \)[/tex] for the given piecewise function
[tex]\[
g(x) = \begin{cases}
x - 1 & \text{if } -2 \leq x < -1 \\
2x + 3 & \text{if } -1 \leq x < 3 \\
6 - x & \text{if } x \geq 3
\end{cases}
\][/tex]
we need to identify which part of the piecewise function applies when [tex]\( x = 3 \)[/tex].
Let's examine the conditions:
1. [tex]\( -2 \leq x < -1 \)[/tex]: This applies when [tex]\( x \)[/tex] is between -2 (inclusive) and -1 (exclusive). It does not apply to [tex]\( x = 3 \)[/tex].
2. [tex]\( -1 \leq x < 3 \)[/tex]: This applies when [tex]\( x \)[/tex] is between -1 (inclusive) and 3 (exclusive). It does not apply to [tex]\( x = 3 \)[/tex].
3. [tex]\( x \geq 3 \)[/tex]: This applies when [tex]\( x \)[/tex] is greater than or equal to 3. Since [tex]\( x = 3 \)[/tex], this condition is met.
The relevant part of the piecewise function for [tex]\( x = 3 \)[/tex] is:
[tex]\[
g(x) = 6 - x
\][/tex]
Substituting [tex]\( x = 3 \)[/tex] into this equation, we get:
[tex]\[
g(3) = 6 - 3 = 3
\][/tex]
Therefore, the value of [tex]\( g(3) \)[/tex] is [tex]\( \boxed{3} \)[/tex].
