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 .

Answer 1

Let's analyze the function based on the description you provided.

1. **At x = -4**: The function has a cusp at the point (-4, 3). A cusp is a point where the function is not continuous, as the left-hand limit and right-hand limit do not match. Therefore, we can conclude:
- The function is **not continuous** at x = -4, and it does not have a removable discontinuity.
- There is a cusp, but it is not classified as a removable discontinuity.

2. **At x = -1**: The function has different behaviors around this point.
- There is an open point at (-1, 0) and a closed point at (-1, 0.75). This indicates a jump because the value of the function jumps from 0.75 to the open value of 0 at x = -1. The function does not take the value of 0 at that point.
- Therefore, we can conclude that there is a jump discontinuity at x = -1.

Considering the analysis above, we can summarize the continuity of the function as follows:
- The function is **not continuous at x = -4** (due to the cusp).
- The function **has a jump discontinuity at x = -1**.

So the correct responses from your options will be:
- The function has a jump discontinuity at x = -1.