Let's find the [tex]\( y \)[/tex]-intercept of the piecewise function given by:
[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]
The [tex]\( y \)[/tex]-intercept is the value of the function [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex].
First, we need to determine which piece of the function to use by checking the domain of each piece:
1. [tex]\(-3x - 2\)[/tex] for [tex]\(-\infty < x < -2\)[/tex]:
- This domain does not include [tex]\( x = 0 \)[/tex].
2. [tex]\(-x + 1\)[/tex] for [tex]\(-2 \leq x < 3\)[/tex]:
- This domain includes [tex]\( x = 0 \)[/tex].
3. [tex]\(2x + 5\)[/tex] for [tex]\(3 \leq x < \infty\)[/tex]:
- This domain does not include [tex]\( x = 0 \)[/tex].
Since [tex]\( x = 0 \)[/tex] falls within the domain [tex]\(-2 \leq x < 3\)[/tex], we use the piece [tex]\(-x + 1\)[/tex] to find the [tex]\( y \)[/tex]-intercept.
Now, evaluate the function at [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = -0 + 1 = 1
\][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept of the function is [tex]\( 1 \)[/tex].
The correct answer is:
A. 1
