Question
A function is graphed on the coordinate plane.The function starts as a line that increasesfrom negative infinity to a cusp at the point left-parenthesis negative 4 comma 3 right-parenthesis. It then turns sharply and decreases as a curve to the point left-parenthesis negative 2 comma negative 1 right-parenthesis. Then it increases to the open point left-parenthesis negative 1 comma 0 right-parenthesis. The function then increases infinitely in a line from the closed point left-parenthesis negative 1 comma 0.75 right-parenthesis.
Question
Use the graph to answer the question.
Describe the continuity of the graphed function. Select all that apply.
(1 point)
Responses
The function is continuous at x equals negative 4.
The function is continuous at Image with alt text: x equals negative 4 .
The function has a removable discontinuity at x equals negative 4.
The function has a removable discontinuity at Image with alt text: x equals negative 4 .
The function has a jump discontinuity at x equals negative 1.
The function has a jump discontinuity at Image with alt text: x equals negative 1 .
The function has an infinite discontinuity at x equals negative 1.
The function has an infinite discontinuity at Image with alt text: x equals negative 1 .
Question
Use the graph to answer the question.
Describe the continuity of the graphed function. Select all that apply.
(1 point)
Responses
The function is continuous at x equals negative 4.
The function is continuous at Image with alt text: x equals negative 4 .
The function has a removable discontinuity at x equals negative 4.
The function has a removable discontinuity at Image with alt text: x equals negative 4 .
The function has a jump discontinuity at x equals negative 1.
The function has a jump discontinuity at Image with alt text: x equals negative 1 .
The function has an infinite discontinuity at x equals negative 1.
The function has an infinite discontinuity at Image with alt text: x equals negative 1 .
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To analyze the continuity of the function as described, we can focus on the specific points of interest: \(x = -4\) and \(x = -1\).
1. **At \(x = -4\)**:
- The function has a cusp at the point \((-4, 3)\). This implies that the behavior of the function changes sharply at this point.
- Therefore, the function is not continuous at \(x = -4\).
- Since there is a sharp change in direction but not a break in the graph, we conclude that there is **not a removable discontinuity**.
- Thus, we can say:
- The function is NOT continuous at \(x = -4\).
- The function does NOT have a removable discontinuity at \(x = -4\).
2. **At \(x = -1\)**:
- The function has an open point at \((-1, 0)\) meaning the point is not included in the function, while it approaches this point from both sides. Additionally, the function is defined at \((-1, 0.75)\) which creates a gap.
- This suggests a jump between the value of the function as it approaches from the left (to 0 as \(x\) approaches \(-1\)) and the value as it approaches or exceeds from the right (continuing along the line above \(0\)).
- Thus, the function displays a jump discontinuity at \(x = -1\) and is not continuous.
Given this analysis, the responses about the continuity of the function are as follows:
- **The function has a jump discontinuity at \(x = -1\)** (This is True).
- **The function does NOT have an infinite discontinuity at \(x = -1\)** (This is False).
So the correct choice regarding the continuity based on your description is:
- **The function has a jump discontinuity at \(x = -1\)**.
1. **At \(x = -4\)**:
- The function has a cusp at the point \((-4, 3)\). This implies that the behavior of the function changes sharply at this point.
- Therefore, the function is not continuous at \(x = -4\).
- Since there is a sharp change in direction but not a break in the graph, we conclude that there is **not a removable discontinuity**.
- Thus, we can say:
- The function is NOT continuous at \(x = -4\).
- The function does NOT have a removable discontinuity at \(x = -4\).
2. **At \(x = -1\)**:
- The function has an open point at \((-1, 0)\) meaning the point is not included in the function, while it approaches this point from both sides. Additionally, the function is defined at \((-1, 0.75)\) which creates a gap.
- This suggests a jump between the value of the function as it approaches from the left (to 0 as \(x\) approaches \(-1\)) and the value as it approaches or exceeds from the right (continuing along the line above \(0\)).
- Thus, the function displays a jump discontinuity at \(x = -1\) and is not continuous.
Given this analysis, the responses about the continuity of the function are as follows:
- **The function has a jump discontinuity at \(x = -1\)** (This is True).
- **The function does NOT have an infinite discontinuity at \(x = -1\)** (This is False).
So the correct choice regarding the continuity based on your description is:
- **The function has a jump discontinuity at \(x = -1\)**.
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