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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.
Feature
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.
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To analyze the continuity or discontinuity of the described function, we'll look at the following aspects based on the graph features you've provided:
1. **First Ray**:
- The first ray starts from negative infinity and increases up to the point \((-1, 2.5)\), and it includes this endpoint (as it is noted to be a solid point).
- There is a hole at the point \((-2, 2)\). This indicates that the function is not defined at this point, which creates a gap in the function.
2. **Isolated Point**:
- There is a single isolated point at \((-2, 1)\). Since this point does not connect to the ray, it further contributes to the discontinuity of the function around \(x = -2\).
3. **Second Ray**:
- The second ray starts at an open point \((-1, 3)\). The open point signifies that this value is not included in the function, which indicates another form of discontinuity at \(x = -1\) since the value jumps from \(2.5\) to \(3\) without including \(3\).
**Conclusion about Continuity/Discontinuity**:
- The function is **discontinuous** at two significant points:
- At \(x = -2\), due to the hole and the isolated point.
- At \(x = -1\), since there is a jump from \(y = 2.5\) (solid point) to \(y = 3\) (open point).
- Therefore, the overall function is discontinuous at these points, indicating gaps or breaks in the graph.
1. **First Ray**:
- The first ray starts from negative infinity and increases up to the point \((-1, 2.5)\), and it includes this endpoint (as it is noted to be a solid point).
- There is a hole at the point \((-2, 2)\). This indicates that the function is not defined at this point, which creates a gap in the function.
2. **Isolated Point**:
- There is a single isolated point at \((-2, 1)\). Since this point does not connect to the ray, it further contributes to the discontinuity of the function around \(x = -2\).
3. **Second Ray**:
- The second ray starts at an open point \((-1, 3)\). The open point signifies that this value is not included in the function, which indicates another form of discontinuity at \(x = -1\) since the value jumps from \(2.5\) to \(3\) without including \(3\).
**Conclusion about Continuity/Discontinuity**:
- The function is **discontinuous** at two significant points:
- At \(x = -2\), due to the hole and the isolated point.
- At \(x = -1\), since there is a jump from \(y = 2.5\) (solid point) to \(y = 3\) (open point).
- Therefore, the overall function is discontinuous at these points, indicating gaps or breaks in the graph.
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