Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Graphing Piecewise Functions
In this exercise, you are asked to graph the piecewise function \( f(x) \) defined as follows:
\[ f(x) = \begin{cases}
3x - 2 & \text{if } x < 2 \\
(x - 2)^2 + 4 & \text{if } 2 \leq x < 3 \\
5 & \text{if } x \geq 3
\end{cases} \]
### Instructions:
1. **Plot the function** for each segment on the coordinate plane.
2. **Identify and label all intercepts** (x-intercepts and y-intercepts).
### Graph Segments Explanation:
- **Segment 1: \( y = 3x - 2 \)**
- This linear function is defined for \( x < 2 \).
- Find the intercepts:
- **Y-intercept** is where \( x = 0 \): \( y = 3(0) - 2 = -2 \).
- **X-intercept** is where \( y = 0 \): \( 0 = 3x - 2 \implies x = \frac{2}{3} \).
- **Segment 2: \( y = (x - 2)^2 + 4 \)**
- This quadratic function is defined for \( 2 \leq x < 3 \).
- Since the domain starts at \( x = 2 \), you need to check the continuity:
- Evaluate at \( x = 2 \): \( y = (2 - 2)^2 + 4 = 4 \).
- Evaluate at \( x = 3 \): \( y = (3 - 2)^2 + 4 = 5 \).
- **Segment 3: \( y = 5 \)**
- This constant function is defined for \( x \geq 3 \).
- It remains constant at \( y = 5 \).
### Instructions for Graphing:
- Use a different color or style of line (solid, dashed) for each segment to clearly differentiate between the piecewise functions.
- Ensure that you include open or closed circles to mark the endpoints of each segment accurately.
- Label the intercepts and any points of interest, especially transitions between the different function definitions.
### Additional Notes](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F432d0c31-f5e9-405a-bdb3-3bc4909f09a2%2Fc01d633b-1f80-4a7e-91e7-f6054211781a%2Fe112r2k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Graphing Piecewise Functions
In this exercise, you are asked to graph the piecewise function \( f(x) \) defined as follows:
\[ f(x) = \begin{cases}
3x - 2 & \text{if } x < 2 \\
(x - 2)^2 + 4 & \text{if } 2 \leq x < 3 \\
5 & \text{if } x \geq 3
\end{cases} \]
### Instructions:
1. **Plot the function** for each segment on the coordinate plane.
2. **Identify and label all intercepts** (x-intercepts and y-intercepts).
### Graph Segments Explanation:
- **Segment 1: \( y = 3x - 2 \)**
- This linear function is defined for \( x < 2 \).
- Find the intercepts:
- **Y-intercept** is where \( x = 0 \): \( y = 3(0) - 2 = -2 \).
- **X-intercept** is where \( y = 0 \): \( 0 = 3x - 2 \implies x = \frac{2}{3} \).
- **Segment 2: \( y = (x - 2)^2 + 4 \)**
- This quadratic function is defined for \( 2 \leq x < 3 \).
- Since the domain starts at \( x = 2 \), you need to check the continuity:
- Evaluate at \( x = 2 \): \( y = (2 - 2)^2 + 4 = 4 \).
- Evaluate at \( x = 3 \): \( y = (3 - 2)^2 + 4 = 5 \).
- **Segment 3: \( y = 5 \)**
- This constant function is defined for \( x \geq 3 \).
- It remains constant at \( y = 5 \).
### Instructions for Graphing:
- Use a different color or style of line (solid, dashed) for each segment to clearly differentiate between the piecewise functions.
- Ensure that you include open or closed circles to mark the endpoints of each segment accurately.
- Label the intercepts and any points of interest, especially transitions between the different function definitions.
### Additional Notes
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