Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Graphing Piecewise Functions
**Instructions:**
Graph the following piecewise function:
\[ f(x) = \begin{cases}
-2 & \text{if } x \leq 2 \\
-1 & \text{if } x > 2
\end{cases} \]
**Choose the correct graph from the options below:**
#### Options:
1. **Option A:**
- This graph shows a line segment where \( f(x) = -2 \) for \( x \leq 2 \) and a horizontal line at \( f(x) = -1 \) for \( x > 2 \).
- It is characterized by a solid dot at the point (2, -2) and an open dot at the point (2, -1), indicating the function changes from -2 to -1 exactly at \( x = 2 \).
2. **Option B:**
- This graph displays a line segment at \( f(x) = -1 \) for \( x \leq 2 \) and a horizontal line at \( f(x) = -2 \) for \( x > 2 \).
- Solid and open dots are swapped, which does not match the conditions given.
3. **Option C:**
- This graph illustrates a line where \( f(x) = -2 \) over all \( x \), which does not match the piecewise function.
4. **Option D:**
- This graph demonstrates a line where \( f(x) = -1 \) for all \( x \), not accurately representing the piecewise nature of the function.
### Solution Explanation:
The correct graph needs to reflect two segments;
- A horizontal line segment at \( f(x) = -2 \) extending up to and including \( x = 2 \).
- Another horizontal line segment at \( f(x) = -1 \) starting just after \( x = 2 \).
Thus:
- The correct graph is **Option A**.
- This graph correctly shows:
- A solid dot at (2, -2), indicating the function's value is -2 at \( x = 2 \).
- An open dot at (2, -1), indicating the function's value immediately changes to -1 for \( x > 2 \).
This detailed explanation should help you understand how to graph piecewise functions and identify the correct](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F599e8835-dd55-414f-94d2-3db99a6d43d5%2F788842a1-c153-40c3-a785-a536a015a3b8%2Fboxxwro_processed.png&w=3840&q=75)
Transcribed Image Text:### Graphing Piecewise Functions
**Instructions:**
Graph the following piecewise function:
\[ f(x) = \begin{cases}
-2 & \text{if } x \leq 2 \\
-1 & \text{if } x > 2
\end{cases} \]
**Choose the correct graph from the options below:**
#### Options:
1. **Option A:**
- This graph shows a line segment where \( f(x) = -2 \) for \( x \leq 2 \) and a horizontal line at \( f(x) = -1 \) for \( x > 2 \).
- It is characterized by a solid dot at the point (2, -2) and an open dot at the point (2, -1), indicating the function changes from -2 to -1 exactly at \( x = 2 \).
2. **Option B:**
- This graph displays a line segment at \( f(x) = -1 \) for \( x \leq 2 \) and a horizontal line at \( f(x) = -2 \) for \( x > 2 \).
- Solid and open dots are swapped, which does not match the conditions given.
3. **Option C:**
- This graph illustrates a line where \( f(x) = -2 \) over all \( x \), which does not match the piecewise function.
4. **Option D:**
- This graph demonstrates a line where \( f(x) = -1 \) for all \( x \), not accurately representing the piecewise nature of the function.
### Solution Explanation:
The correct graph needs to reflect two segments;
- A horizontal line segment at \( f(x) = -2 \) extending up to and including \( x = 2 \).
- Another horizontal line segment at \( f(x) = -1 \) starting just after \( x = 2 \).
Thus:
- The correct graph is **Option A**.
- This graph correctly shows:
- A solid dot at (2, -2), indicating the function's value is -2 at \( x = 2 \).
- An open dot at (2, -1), indicating the function's value immediately changes to -1 for \( x > 2 \).
This detailed explanation should help you understand how to graph piecewise functions and identify the correct
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