To determine which graph represents the given function [tex]\( f(x) \)[/tex], we need to analyze the behavior and the values of [tex]\( f(x) \)[/tex] for different intervals of [tex]\( x \)[/tex].
Given,
[tex]\[
f(x)=\left\{\begin{array}{ll}
5, & x<-2 \\
3, & -2 \leq x<0 \\
0, & 0 \leq x<2 \\
-3, & x \geq 2
\end{array}\right.
\][/tex]
Let's break down the function step-by-step:
1. For [tex]\( x < -2 \)[/tex]:
[tex]\( f(x) = 5 \)[/tex].
This part is represented by a horizontal line at [tex]\( y = 5 \)[/tex] which continues until [tex]\( x = -2 \)[/tex], but does not include [tex]\( x = -2 \)[/tex].
2. For [tex]\( -2 \leq x < 0 \)[/tex]:
[tex]\( f(x) = 3 \)[/tex].
This part is represented by a horizontal line at [tex]\( y = 3 \)[/tex] which starts at [tex]\( x = -2 \)[/tex] and ends just before [tex]\( x = 0 \)[/tex]. There is a closed circle at [tex]\( x = -2 \)[/tex] and an open circle at [tex]\( x = 0 \)[/tex].
3. For [tex]\( 0 \leq x < 2 \)[/tex]:
[tex]\( f(x) = 0 \)[/tex].
This part is represented by a horizontal line at [tex]\( y = 0 \)[/tex] which starts at [tex]\( x = 0 \)[/tex] and ends just before [tex]\( x = 2 \)[/tex]. There is a closed circle at [tex]\( x = 0 \)[/tex] and an open circle at [tex]\( x = 2 \)[/tex].
4. For [tex]\( x \geq 2 \)[/tex]:
[tex]\( f(x) = -3 \)[/tex].
This part is represented by a horizontal line at [tex]\( y = -3 \)[/tex] which starts from [tex]\( x = 2 \)[/tex] and continues indefinitely. There is a closed circle at [tex]\( x = 2 \)[/tex].
Let's match these observations with the given graph options:
- We need a graph that has a horizontal line at [tex]\( y = 5 \)[/tex] for [tex]\( x < -2 \)[/tex].
- It should have a horizontal line at [tex]\( y = 3 \)[/tex] from [tex]\( x = -2 \)[/tex] to just before [tex]\( x = 0 \)[/tex] with a closed circle at [tex]\( x = -2 \)[/tex] and an open circle at [tex]\( x = 0 \)[/tex].
- There should be a horizontal line at [tex]\( y = 0 \)[/tex] from [tex]\( x = 0 \)[/tex] to just before [tex]\( x = 2 \)[/tex] with a closed circle at [tex]\( x = 0 \)[/tex] and an open circle at [tex]\( x = 2 \)[/tex].
- Finally, there should be a horizontal line at [tex]\( y = -3 \)[/tex] from [tex]\( x = 2 \)[/tex] onwards with a closed circle at [tex]\( x = 2 \)[/tex].
By comparing these criteria with the provided graphs, you will be able to identify the graph that correctly represents the function [tex]\( f(x) \)[/tex].
