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Let's analyze the graph to determine the types of discontinuities at each \( x \)-value.

1. **At \( x = -4 \)**: 
   - From the graph, it appears that the function has a **jump discontinuity** at \( x = -4 \). There are two distinct parts of the graph on either side of \( x = -4 \), indicating that the left-hand and right-hand limits do not match.

2. **At \( x = -2 \)**: 
   - At this point, the graph shows a **removable discontinuity** because there is a hole at \( x = -2 \), but the graph approaches the same value from both sides. The function is not defined at this point, but the limits from the left and right are equal.

3. **At \( x = 2 \)**: 
   - The graph suggests a **removable discontinuity** here as well. There is a hole at \( x = 2 \), but the function could be redefined to fill the hole, as the left-hand and right-hand limits match.

4. **At \( x = 4 \)**: 
   - The graph seems to exhibit a **jump discontinuity** at \( x = 4 \), similar to the case at \( x = -4 \). The left-hand and right-hand limits do not approach the same value, and there is a jump between two parts of the function.

If you'd like more details on any of these, feel free to ask!

Here are some follow-up questions for deeper understanding:
1. How can you formally define the different types of discontinuities?
2. What is the difference between a removable discontinuity and a jump discontinuity?
3. Can a function be continuous at a point where it has a removable discontinuity if redefined?
4. How do limits from the left-hand side and right-hand side relate to determining the type of discontinuity?
5. How would you find discontinuities algebraically, rather than just graphically?

**Tip**: Always check if the left-hand and right-hand limits are equal when identifying discontinuities!

Use the given graph of the function f to determine the type of discontinuity at each x-value.

Learn how to determine the types of discontinuities, including jump, removable, and infinite discontinuities, by analyzing the graph of a function. This problem focuses on limits and the behavior of functions at various points. Suitable for students in Grades 11-12.

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