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 analyze the continuity or discontinuity of the function described, we can break down the information given:
1. **First Ray:**
- It increases from negative infinity to the point \((-1, 2.5)\), and this endpoint is included in the function, which means the function is continuous up to this point.
- There is a hole at \((-2, 2)\), which signifies a discontinuity since the function is not defined at this point. This means that the limit of the function as it approaches \(-2\) does not equal the function value at that point.
2. **Isolated Point:**
- There is an isolated point at \((-2, 1)\). This point does not connect to the hole at \((-2, 2)\), indicating that at \(-2\), the function jumps from not existing at the hole to being \(1\) at the isolated point. This creates a discontinuity as well.
3. **Second Ray:**
- The second ray starts at an open point at \((-1, 3)\) and decreases to infinity. The open point means that the function is not defined at \((-1, 3)\), which contributes to the discontinuity since the function does not approach this point from the left (it comes from the ray).
### Summary of Continuity or Discontinuity:
- The function has a **discontinuity at \(-2\)** due to the hole and the isolated point.
- The function is also **discontinuous at \(-1\)** because of the open point there.
- The transition from the first ray to the second ray creates additional points of discontinuity at \(-1\).
### Final Description:
The graphed function is discontinuous at \(-2\) (due to the hole and isolated point) and at \(-1\) (due to the open point), while being continuous on the intervals defined by those points outside these discontinuities. Overall, the function exhibits a jump discontinuity at both \(-2\) and \(-1\).
1. **First Ray:**
- It increases from negative infinity to the point \((-1, 2.5)\), and this endpoint is included in the function, which means the function is continuous up to this point.
- There is a hole at \((-2, 2)\), which signifies a discontinuity since the function is not defined at this point. This means that the limit of the function as it approaches \(-2\) does not equal the function value at that point.
2. **Isolated Point:**
- There is an isolated point at \((-2, 1)\). This point does not connect to the hole at \((-2, 2)\), indicating that at \(-2\), the function jumps from not existing at the hole to being \(1\) at the isolated point. This creates a discontinuity as well.
3. **Second Ray:**
- The second ray starts at an open point at \((-1, 3)\) and decreases to infinity. The open point means that the function is not defined at \((-1, 3)\), which contributes to the discontinuity since the function does not approach this point from the left (it comes from the ray).
### Summary of Continuity or Discontinuity:
- The function has a **discontinuity at \(-2\)** due to the hole and the isolated point.
- The function is also **discontinuous at \(-1\)** because of the open point there.
- The transition from the first ray to the second ray creates additional points of discontinuity at \(-1\).
### Final Description:
The graphed function is discontinuous at \(-2\) (due to the hole and isolated point) and at \(-1\) (due to the open point), while being continuous on the intervals defined by those points outside these discontinuities. Overall, the function exhibits a jump discontinuity at both \(-2\) and \(-1\).