x 2) Discuss the continuity of the graph of f(x) given below on the closed interval [-4, 2] D(-4,6) (2,3.5) (-1,2) (2,1) (-4,4) (-1,-1)² (2,2) X
x 2) Discuss the continuity of the graph of f(x) given below on the closed interval [-4, 2] D(-4,6) (2,3.5) (-1,2) (2,1) (-4,4) (-1,-1)² (2,2) X
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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![**Title: Discussing Continuity of $f(x)$ on the Closed Interval [-4, 2]**
In this section, we explore the continuity of the function \( f(x) \) over the interval \([-4, 2]\). The graph provided visualizes the behavior of \( f(x) \) and displays several key points and features to consider when discussing continuity.
### Graph Explanation:
- **Points and Intervals of Interest:**
- The graph is defined on the interval \([-4, 2]\), which includes several specific points with both solid and open dots indicating the behavior of \( f(x) \) at these points.
- **Key Points:**
- \((-4, 6)\) - Open circle indicating \( f(-4) \) is not defined at 6.
- \((-4, 4)\) - Solid dot indicating \( f(-4) = 4 \).
- \((-1, 2)\) - Solid dot indicating \( f(-1) = 2 \).
- \((-1, -1)\) - Solid dot indicating \( f(-1) = -1 \).
- \((2, 3.5)\) - Solid dot indicating \( f(2) = 3.5 \).
- \((2, 2)\) - Open circle indicating \( f(2) \) is not defined at 2.
- \((2, 1)\) - Open circle indicating \( f(2) \) is not defined at 1.
### Continuity Discussion:
1. **Intervals Where \( f(x) \) is Continuous:**
- The function appears to be continuous between \((-4, 4)\) and \((-1, 2)\), as these segments display a connected path without interruptions.
2. **Discontinuities:**
- **At \( x = -4 \):** There is a jump discontinuity where the value of the function jumps from 6 to 4.
- **At \( x = -1 \):** There is an issue as there are two values, \((-1, 2)\) and \((-1, -1)\), indicating a discontinuity. It implies that the function is not single-valued at \( x = -1 \).
- **At \( x = 2 \):**
- There is an](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80a2c7ea-75cb-4270-a574-afc900ab9545%2Fa5490c33-e5f1-4da8-8c55-5cf22db8072e%2Fr2f8mon_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Discussing Continuity of $f(x)$ on the Closed Interval [-4, 2]**
In this section, we explore the continuity of the function \( f(x) \) over the interval \([-4, 2]\). The graph provided visualizes the behavior of \( f(x) \) and displays several key points and features to consider when discussing continuity.
### Graph Explanation:
- **Points and Intervals of Interest:**
- The graph is defined on the interval \([-4, 2]\), which includes several specific points with both solid and open dots indicating the behavior of \( f(x) \) at these points.
- **Key Points:**
- \((-4, 6)\) - Open circle indicating \( f(-4) \) is not defined at 6.
- \((-4, 4)\) - Solid dot indicating \( f(-4) = 4 \).
- \((-1, 2)\) - Solid dot indicating \( f(-1) = 2 \).
- \((-1, -1)\) - Solid dot indicating \( f(-1) = -1 \).
- \((2, 3.5)\) - Solid dot indicating \( f(2) = 3.5 \).
- \((2, 2)\) - Open circle indicating \( f(2) \) is not defined at 2.
- \((2, 1)\) - Open circle indicating \( f(2) \) is not defined at 1.
### Continuity Discussion:
1. **Intervals Where \( f(x) \) is Continuous:**
- The function appears to be continuous between \((-4, 4)\) and \((-1, 2)\), as these segments display a connected path without interruptions.
2. **Discontinuities:**
- **At \( x = -4 \):** There is a jump discontinuity where the value of the function jumps from 6 to 4.
- **At \( x = -1 \):** There is an issue as there are two values, \((-1, 2)\) and \((-1, -1)\), indicating a discontinuity. It implies that the function is not single-valued at \( x = -1 \).
- **At \( x = 2 \):**
- There is an
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