To determine the [tex]\( y \)[/tex]-intercept of the function [tex]\( f(x) \)[/tex], we need to evaluate the function at [tex]\( x = 0 \)[/tex]. The [tex]\( y \)[/tex]-intercept is the value of [tex]\( y \)[/tex] where the graph of the function crosses the [tex]\( y \)[/tex]-axis.
The given piecewise function [tex]\( f(x) \)[/tex] is defined as:
[tex]\[
f(x) =
\begin{cases}
-3x - 2, & \text{for } -\infty < x < -2 \\
-x + 1, & \text{for } -2 \leq x < 3 \\
2x + 5, & \text{for } 3 \leq x < \infty
\end{cases}
\][/tex]
Let's check which piece of the function is valid at [tex]\( x = 0 \)[/tex]. The condition [tex]\( -2 \leq x < 3 \)[/tex] includes [tex]\( x = 0 \)[/tex]. Thus, we use the second piece of the function ([tex]\( f(x) = -x + 1 \)[/tex]) to find the [tex]\( y \)[/tex]-intercept.
Now, substitute [tex]\( x = 0 \)[/tex] into the second piece:
[tex]\[
f(0) = -0 + 1 = 1
\][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept of the function [tex]\( f(x) \)[/tex] is [tex]\( 1 \)[/tex].
The correct answer is [tex]\( \boxed{1} \)[/tex].
