https://file.aiquickdraw.com/mathbot/file/1726798969748fp2t02dx.jpg

https://file.aiquickdraw.com/mathbot/file/1726798969749do8e1rk2.jpg

From the graph in the first image, we can identify the function's behavior and locate the discontinuities at \( x = -4 \), \( x = -2 \), \( x = 2 \), and \( x = 4 \). I will now describe the types of discontinuities at each of these points based on the graph:

1. **At \( x = -4 \)**:  
   The function seems to have a **removable discontinuity**. There is a hole (open circle) at this point, indicating the limit exists, but the function is not defined at \( x = -4 \).

2. **At \( x = -2 \)**:  
   There is a **jump discontinuity**. The graph breaks, with a gap between two segments, and the function takes two different values as you approach from the left and the right.

3. **At \( x = 2 \)**:  
   Another **removable discontinuity** occurs here. The open circle indicates the function has a hole, suggesting the limit exists, but the value is undefined or different at this point.

4. **At \( x = 4 \)**:  
   The function has an **infinite discontinuity**. The graph shoots upwards on one side and downwards on the other, meaning the function approaches infinity (vertical asymptote) as \( x \) approaches 4 from either side.

Let me know if you'd like more detailed explanations or have further questions.

Here are 5 follow-up questions to expand your understanding:
1. How do you calculate limits approaching a point of removable discontinuity?
2. What is the difference between a jump and infinite discontinuity?
3. Why is the function's behavior different at a removable discontinuity compared to a jump discontinuity?
4. Can a function be continuous at a point with a removable discontinuity if we redefine the value at that point?
5. What types of real-world phenomena can be modeled with functions that have discontinuities?

**Tip:** Removable discontinuities can often be "fixed" by redefining the function at that point, making the function continuous.

What type of discontinuity does f have at x = -4, x = -2, x = 2, and x = 4?

Explore the types of discontinuities in a function, including removable, jump, and infinite discontinuities, as seen on a graph at points x = -4, -2, 2, and 4. Understand how limits and vertical asymptotes are involved in discontinuous behavior. Suitable for students in Grades 11-12 studying precalculus or calculus.

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