Sketch a graph of f(z) -4-3 13 if -2z-1 if -1 z ≤-1 -1 < z <1 if z > 1 Clear All Draw: Line Dot Open Dot Notes: Only use open/closed dots if two parts of the function don't connect. To draw a line segment click once at one end, then move mouse to other end and click again. Then, click outside the graph if you need to end the line.
Sketch a graph of f(z) -4-3 13 if -2z-1 if -1 z ≤-1 -1 < z <1 if z > 1 Clear All Draw: Line Dot Open Dot Notes: Only use open/closed dots if two parts of the function don't connect. To draw a line segment click once at one end, then move mouse to other end and click again. Then, click outside the graph if you need to end the line.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Piecewise Functions - Graphing Example
#### Problem Statement:
Sketch a graph of \( f(x) \):
\[
f(x) =
\begin{cases}
1 & \text{if} \ x \leq -1 \\
-2x - 1 & \text{if} \ -1 < x \leq 1 \\
-1 & \text{if} \ x > 1
\end{cases}
\]
#### Graph Explanation:
The given graph consists of a coordinate grid with the x-axis and y-axis both ranging from -5 to 5. The piecewise function defined above is plotted on this grid with the following details:
1. **For \( x \leq -1 \)**: The function is constant at \( y = 1 \). This means for all x-values less than or equal to -1, the y-value is 1. The segment is closed and horizontal extending infinitely to the left.
2. **For \( -1 < x \leq 1 \)**: The function is linear and given by \( y = -2x - 1 \). In this domain, the line starts just to the right of \( x = -1 \), where y would be slightly greater than \( y = 1 \), and continues until \( x = 1 \) (inclusive), where \( y = -3 \). The graph shows this as a diagonal line segment connecting the points that correspond to \( x = -1 \) and \( x = 1 \).
3. **For \( x > 1 \)**: Again, the function is constant but this time at \( y = -1 \). This segment is horizontal and closed, extending infinitely to the right from \( x = 1 \).
#### Notes:
- **Open/Closed Dots**: Use open or closed dots to indicate whether the endpoints of segments are included or not. Closed dots mean that the point is included (≤ or ≥), while an open dot means that the point is not included (< or >).
- **Drawing Instructions**: Click once at one end of the line segment, then move the mouse to the other endpoint and click again to draw a line segment. Click outside the graph if you need to end the line.
#### Additional Resources:
- For further assistance on drawing piecewise functions, refer to the following instructional videos:
* [Video 1](#)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6ef2e86f-df5b-40a3-8009-016cf0671dd4%2F504aa76d-5bbf-4399-a5d6-edb07b1b1810%2Fwovcj9f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Piecewise Functions - Graphing Example
#### Problem Statement:
Sketch a graph of \( f(x) \):
\[
f(x) =
\begin{cases}
1 & \text{if} \ x \leq -1 \\
-2x - 1 & \text{if} \ -1 < x \leq 1 \\
-1 & \text{if} \ x > 1
\end{cases}
\]
#### Graph Explanation:
The given graph consists of a coordinate grid with the x-axis and y-axis both ranging from -5 to 5. The piecewise function defined above is plotted on this grid with the following details:
1. **For \( x \leq -1 \)**: The function is constant at \( y = 1 \). This means for all x-values less than or equal to -1, the y-value is 1. The segment is closed and horizontal extending infinitely to the left.
2. **For \( -1 < x \leq 1 \)**: The function is linear and given by \( y = -2x - 1 \). In this domain, the line starts just to the right of \( x = -1 \), where y would be slightly greater than \( y = 1 \), and continues until \( x = 1 \) (inclusive), where \( y = -3 \). The graph shows this as a diagonal line segment connecting the points that correspond to \( x = -1 \) and \( x = 1 \).
3. **For \( x > 1 \)**: Again, the function is constant but this time at \( y = -1 \). This segment is horizontal and closed, extending infinitely to the right from \( x = 1 \).
#### Notes:
- **Open/Closed Dots**: Use open or closed dots to indicate whether the endpoints of segments are included or not. Closed dots mean that the point is included (≤ or ≥), while an open dot means that the point is not included (< or >).
- **Drawing Instructions**: Click once at one end of the line segment, then move the mouse to the other endpoint and click again to draw a line segment. Click outside the graph if you need to end the line.
#### Additional Resources:
- For further assistance on drawing piecewise functions, refer to the following instructional videos:
* [Video 1](#)
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