Question
A function consisting of two rays and an isolated point is graphed on a coordinate plane.A function consisting of two rays and an isolated point is graphed on a coordinate plane. The first ray increases from negative infinity to the point left parenthesis negative 1 comma 2.5 right parenthesis and includes the endpoint. The ray also has a hole at left parenthesis negative 2 comma 2 right parenthesis. A single point is graphed at left parenthesis negative 2 comma 1 right parenthesis. The second ray starts at an open point at left parenthesis negative 1 comma 3 right parenthesis and decreases to infinity.
Question
Use the graph to answer the question.
Describe the continuity or discontinuity of the graphed function.
(4 points)
Question
Use the graph to answer the question.
Describe the continuity or discontinuity of the graphed function.
(4 points)
Answers
GPT-4o mini
To describe the continuity or discontinuity of the graphed function, we need to analyze the information given about the rays and isolated points.
1. **First Ray**: This ray increases and includes the endpoint at \((-1, 2.5)\) but has a hole at \((-2, 2)\). This means that as you approach \((-2, 2)\) from the left on the first ray, the function approaches \(2\), but it is not defined at that point (hole). Therefore, there is a point of discontinuity at \((-2, 2)\).
2. **Isolated Point**: There is a single point graphed at \((-2, 1)\). This point does not affect the continuity of the function at \((-2, 2)\) since the function was originally discontinuous at that location. The isolated point does not connect with the rays, further emphasizing that the function is not continuous at \(x = -2\).
3. **Second Ray**: The second ray starts at an open point at \((-1, 3)\) and decreases to infinity. The openness at \((-1, 3)\) indicates another point of discontinuity at \((-1, 3)\) since the function does not actually attain this value at \(x = -1\). It transitions from the first ray to this open point, suggesting a jump discontinuity.
### Summary:
- There is a **discontinuity at** \((-2, 2)\) due to the hole.
- There is another **discontinuity at** \((-1, 3)\) due to the open point.
- The function is generally **discontinuous** at these points, and while it is continuous on the intervals outside these points, the specified points cause overall discontinuity in the function.
**Overall conclusion**: The graphed function has points of discontinuity at \((-2, 2)\) and \((-1, 3)\), making the function discontinuous.
1. **First Ray**: This ray increases and includes the endpoint at \((-1, 2.5)\) but has a hole at \((-2, 2)\). This means that as you approach \((-2, 2)\) from the left on the first ray, the function approaches \(2\), but it is not defined at that point (hole). Therefore, there is a point of discontinuity at \((-2, 2)\).
2. **Isolated Point**: There is a single point graphed at \((-2, 1)\). This point does not affect the continuity of the function at \((-2, 2)\) since the function was originally discontinuous at that location. The isolated point does not connect with the rays, further emphasizing that the function is not continuous at \(x = -2\).
3. **Second Ray**: The second ray starts at an open point at \((-1, 3)\) and decreases to infinity. The openness at \((-1, 3)\) indicates another point of discontinuity at \((-1, 3)\) since the function does not actually attain this value at \(x = -1\). It transitions from the first ray to this open point, suggesting a jump discontinuity.
### Summary:
- There is a **discontinuity at** \((-2, 2)\) due to the hole.
- There is another **discontinuity at** \((-1, 3)\) due to the open point.
- The function is generally **discontinuous** at these points, and while it is continuous on the intervals outside these points, the specified points cause overall discontinuity in the function.
**Overall conclusion**: The graphed function has points of discontinuity at \((-2, 2)\) and \((-1, 3)\), making the function discontinuous.