Certainly! Let's break down the process step by step.
1. Understand the Piecewise Function:
The function [tex]\( f(x) \)[/tex] is defined as follows:
[tex]\[
f(x) = \begin{cases}
2 & \text{if } -5 < x \leq -1 \\
4 & \text{if } -1 < x \leq 2 \\
6 & \text{if } 2 < x \leq 5
\end{cases}
\][/tex]
2. Identify the Domains and Ranges:
- For the first interval [tex]\(-5 < x \leq -1\)[/tex], the function value is 2.
- For the second interval [tex]\(-1 < x \leq 2\)[/tex], the function value is 4.
- For the third interval [tex]\(2 < x \leq 5\)[/tex], the function value is 6.
3. Plot the Intervals on the Graph:
- For [tex]\(-5 < x \leq -1\)[/tex], draw a horizontal line at [tex]\(y = 2\)[/tex]. Be sure to use an open circle at [tex]\(x = -5\)[/tex] and a closed circle at [tex]\(x = -1\)[/tex].
- For [tex]\(-1 < x \leq 2\)[/tex], draw a horizontal line at [tex]\(y = 4\)[/tex]. Use an open circle at [tex]\(x = -1\)[/tex] and a closed circle at [tex]\(x = 2\)[/tex].
- For [tex]\(2 < x \leq 5\)[/tex], draw a horizontal line at [tex]\(y = 6\)[/tex]. Use an open circle at [tex]\(x = 2\)[/tex] and a closed circle at [tex]\(x = 5\)[/tex].
4. Ensure Continuous intervals:
- The function should be a series of horizontal line segments without vertical jumps at the transition points.
5. Match with the Options Given:
Since we are plotting the following:
- A horizontal line at [tex]\(y = 2\)[/tex] from [tex]\(-5\)[/tex] (open circle) to [tex]\(-1\)[/tex] (closed circle).
- A horizontal line at [tex]\(y = 4\)[/tex] from [tex]\(-1\)[/tex] (open circle) to [tex]\(2\)[/tex] (closed circle).
- A horizontal line at [tex]\(y = 6\)[/tex] from [tex]\(2\)[/tex] (open circle) to [tex]\(5\)[/tex] (closed circle).
Look through each of the answer choices—A, B, and C—and see which one matches this description.
Without access to the exact descriptions for A, B, and C, you will need to carefully consider each option's text description to see which one accurately corresponds to the graph you have drawn. The correct answer will precisely describe the intervals and their respective function values as horizontally segmented lines, transitioning correctly at -1 and 2.
If you have any further details about the text descriptions in options A, B, and C, feel free to compare and match them to ensure accuracy!
