To find the [tex]\( y \)[/tex]-intercept of the function [tex]\( f(x) \)[/tex], we need to determine the [tex]\( y \)[/tex]-value when [tex]\( x = 0 \)[/tex]. The given piecewise function is:
[tex]\[
f(x)=\left\{\begin{array}{ll}
-3 x-2, & -\infty < x < -2 \\
-x+1, & -2 \leq x < 3 \\
2 x + 5, & 3 \leq x < \infty
\end{array}\right.
\][/tex]
Let's evaluate in which interval [tex]\( x = 0 \)[/tex] falls. The interval definitions are as follows:
1. [tex]\( -\infty < x < -2 \)[/tex]
2. [tex]\( -2 \leq x < 3 \)[/tex]
3. [tex]\( 3 \leq x < \infty \)[/tex]
Since [tex]\( 0 \)[/tex] is in the interval [tex]\( -2 \leq x < 3 \)[/tex], we use the corresponding function segment [tex]\( f(x) = -x + 1 \)[/tex].
Next, substitute [tex]\( x = 0 \)[/tex] into the segment function:
[tex]\[
f(0) = -0 + 1
\][/tex]
[tex]\[
f(0) = 1
\][/tex]
So the [tex]\( y \)[/tex]-intercept of the function [tex]\( f(x) \)[/tex] is [tex]\( 1 \)[/tex]. Hence, the correct answer is:
A. 1
