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What is the [tex]\( y \)[/tex]-intercept of the function [tex]\( f \)[/tex]?
[tex]\[
f(x) = \left\{
\begin{array}{ll}
-3x - 2, & -\infty \ \textless \ x \ \textless \ -2 \\
-x + 1, & -2 \leq x \ \textless \ 3 \\
2x + 5, & 3 \leq x \ \textless \ \infty
\end{array}
\right.
\][/tex]
A. 1
B. -1
C. -2
D. 5



Answer :

GinnyAnswer
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

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