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Question

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.

Answer 1

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.