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State the intervals on which f(x) is continuous. (Enter your answer using interval notation.)


To determine the intervals where \( f(x) \) is continuous based on the graph, let's examine the function:

- **Continuity requires that the graph has no breaks, jumps, or holes.**
- Based on the graph provided, there are no breaks between \( x = -7 \) and \( x = 4 \) except at two points:
  1. There is a hole at \( x = -6 \), indicating discontinuity there.
  2. At \( x = 2 \), the graph has a sharp break, indicating discontinuity as well.

### Conclusion:

- The function is continuous in the following intervals (excluding the points of discontinuity at \( x = -6 \) and \( x = 2 \)):
  \[
  (-7, -6) \cup (-6, 2) \cup (2, 4]
  \]

Would you like further details on this solution, or do you have any other questions?

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Here are 5 related questions to further your understanding:

1. What types of discontinuities exist, and how do they differ?
2. How can you tell if a function has a removable discontinuity from its graph?
3. How do we formally define continuity at a point using limits?
4. How would the intervals change if the graph had a vertical asymptote?
5. How can you determine whether a point of discontinuity is removable or non-removable?

**Tip:** When checking continuity from a graph, always look for holes, jumps, and vertical asymptotes, as these indicate points where the function is not continuous.

State the intervals on which f(x) is continuous. (Enter your answer using interval notation.)

Explore how to determine the intervals where a function is continuous based on its graph. Learn how to identify points of discontinuity and use interval notation to describe continuity for the function on the interval [-7, 4]. Suitable for high school calculus students.

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