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The graph of y f(x) is shown. Use the graph to answer the question. 7) Is f continuous at x = -1? IM 5. -4 -3.-2.-1

The graph of y f(x) is shown. Use the graph to answer the question. 7) Is f continuous at x = -1? IM 5. -4 -3.-2.-1

Calculus: Early Transcendentals
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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**Understanding Continuity from a Graph** **Question on Continuity:** The graph of \( y = f(x) \) is shown. Use the graph to answer the question. 7) Is \( f \) continuous at \( x = -1 \)? **Detailed Description of the Provided Graph:** The graph provided depicts a piecewise function. - There is a horizontal line segment extending from \( x = -5 \) to \( x = -1 \) with an open circle at \( x = -1 \) and a filled circle at \( y = 3 \). - There is a point at \( (-1, -3) \). - From \( x = -1 \), there is an upward-sloping line extending through the point \( (0, 0) \) and continuing to \( (5, 4) \). **Explanation of Continuity at \( x = -1 \):** To determine if the function is continuous at \( x = -1 \), we need to check three conditions: 1. The function \( f \) must be defined at \( x = -1 \). 2. The limit of \( f(x) \) as \( x \) approaches \(-1\) from the left (\( \lim_{{x \to -1^-}} f(x) \)) must exist. 3. The limit of \( f(x) \) as \( x \) approaches \(-1\) from the right (\( \lim_{{x \to -1^+}} f(x) \)) must exist. 4. The limit from the left and the right must be equal, and the function value at \( x = -1 \) must match this limit. From the graph: - \( f(-1) = -3 \) as indicated by the filled circle at \( (-1, -3) \). - The limit from the left as \( x \) approaches \(-1\) is at \( y = 3 \), where the open circle indicates that the value just before \( x = -1 \) is 3. - The limit from the right as \( x \) approaches \(-1\) follows the line passing through points \( (0,0) \) and \( (5,4) \), indicating it also leads to a different \( y \)-value approaching from the right but not touching the point directly at \( x = -
Transcribed Image Text:**Understanding Continuity from a Graph** **Question on Continuity:** The graph of \( y = f(x) \) is shown. Use the graph to answer the question. 7) Is \( f \) continuous at \( x = -1 \)? **Detailed Description of the Provided Graph:** The graph provided depicts a piecewise function. - There is a horizontal line segment extending from \( x = -5 \) to \( x = -1 \) with an open circle at \( x = -1 \) and a filled circle at \( y = 3 \). - There is a point at \( (-1, -3) \). - From \( x = -1 \), there is an upward-sloping line extending through the point \( (0, 0) \) and continuing to \( (5, 4) \). **Explanation of Continuity at \( x = -1 \):** To determine if the function is continuous at \( x = -1 \), we need to check three conditions: 1. The function \( f \) must be defined at \( x = -1 \). 2. The limit of \( f(x) \) as \( x \) approaches \(-1\) from the left (\( \lim_{{x \to -1^-}} f(x) \)) must exist. 3. The limit of \( f(x) \) as \( x \) approaches \(-1\) from the right (\( \lim_{{x \to -1^+}} f(x) \)) must exist. 4. The limit from the left and the right must be equal, and the function value at \( x = -1 \) must match this limit. From the graph: - \( f(-1) = -3 \) as indicated by the filled circle at \( (-1, -3) \). - The limit from the left as \( x \) approaches \(-1\) is at \( y = 3 \), where the open circle indicates that the value just before \( x = -1 \) is 3. - The limit from the right as \( x \) approaches \(-1\) follows the line passing through points \( (0,0) \) and \( (5,4) \), indicating it also leads to a different \( y \)-value approaching from the right but not touching the point directly at \( x = -
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