To determine which graph represents the given function [tex]\( f(x) \)[/tex], we need to break down the function piece by piece according to the given ranges of [tex]\( x \)[/tex]. We will then visually match these pieces to the correct graph.
The function [tex]\( f(x) \)[/tex] is defined as follows:
[tex]\[
f(x) = \left\{
\begin{array}{ll}
5, & \text{for } x < -2 \\
3, & \text{for } -2 \leq x < 0 \\
0, & \text{for } 0 \leq x < 2 \\
-3, & \text{for } x \geq 2
\end{array}
\right.
\][/tex]
Let's analyze each piece of the function:
1. For [tex]\( x < -2 \)[/tex], [tex]\( f(x) = 5 \)[/tex]:
- This means for all [tex]\( x \)[/tex] values less than -2, the function has a constant value of 5.
2. For [tex]\( -2 \leq x < 0 \)[/tex], [tex]\( f(x) = 3 \)[/tex]:
- This means from [tex]\( x = -2 \)[/tex] to just less than 0 (but not including 0), the function takes on a constant value of 3.
3. For [tex]\( 0 \leq x < 2 \)[/tex], [tex]\( f(x) = 0 \)[/tex]:
- This means from [tex]\( x = 0 \)[/tex] to just less than 2 (but not including 2), the function takes on a constant value of 0.
4. For [tex]\( x \geq 2 \)[/tex], [tex]\( f(x) = -3 \)[/tex]:
- This means for all [tex]\( x \)[/tex] values greater than or equal to 2, the function has a constant value of -3.
To visualize this, let's describe what the graph should look like:
- A horizontal line at [tex]\( y = 5 \)[/tex] for [tex]\( x < -2 \)[/tex].
- A horizontal line at [tex]\( y = 3 \)[/tex] for [tex]\( -2 \leq x < 0 \)[/tex], including a closed circle at [tex]\( x = -2 \)[/tex] and an open circle just before [tex]\( x = 0 \)[/tex].
- A horizontal line at [tex]\( y = 0 \)[/tex] for [tex]\( 0 \leq x < 2 \)[/tex], including a closed circle at [tex]\( x = 0 \)[/tex] and an open circle just before [tex]\( x = 2 \)[/tex].
- A horizontal line at [tex]\( y = -3 \)[/tex] for [tex]\( x \geq 2 \)[/tex], including a closed circle at [tex]\( x = 2 \)[/tex].
Now, look at the graphs provided and match the correct graph with these criteria:
- Graph [tex]\( A \)[/tex] or Graph [tex]\( B \)[/tex].
Based on the description, the correct graph should visually meet all the above criteria. Therefore, the correct answer is [tex]\( \boxed{A} \)[/tex].
