https://file.aiquickdraw.com/mathbot/file/1737603642683ji6qjq3l.jpg

https://file.aiquickdraw.com/mathbot/file/1737603642684trc7ty1i.jpg

The two images you uploaded depict a piecewise function and its graphical representation. Based on the function and graph, here’s how we can analyze the categorization:

1. **Positive/Negative Intervals:**
   - **Positive** intervals refer to regions where the function \( f(x) \) is above the x-axis.
   - **Negative** intervals refer to regions where the function \( f(x) \) is below the x-axis.

   Looking at the graph:
   - The function is positive for \( (0, \infty) \).
   - The function is negative for \( (-\infty, 0) \).

2. **Increasing/Decreasing Intervals:**
   - **Increasing** intervals refer to parts of the graph where \( f(x) \) is going upward as \( x \) increases.
   - **Decreasing** intervals refer to parts of the graph where \( f(x) \) is going downward as \( x \) increases.

   Based on the graph:
   - The function is **increasing** for \( (1, \infty) \) (where the graph is rising).
   - The function is **decreasing** for \( (-\infty, 0) \) (where the graph is falling).

**Answer Summary:**

1. **Positive Intervals:** \( (0, \infty) \)
   **Negative Intervals:** \( (-\infty, 0) \)
   
2. **Increasing Intervals:** \( (1, \infty) \)
   **Decreasing Intervals:** \( (-\infty, 0) \)

Would you like me to clarify anything or provide further details on these intervals? 

### Related Questions:
1. How would you define a piecewise function mathematically?
2. Can you explain how to calculate the slope of a function over different intervals?
3. What are some common types of piecewise functions that you encounter in real-life applications?
4. How do you determine if a function is continuous or discontinuous from its graph?
5. How do you determine the domain and range of a piecewise function?

**Tip:** When identifying increasing and decreasing intervals, always examine the slope of the graph—if it’s rising, it’s increasing, and if it’s falling, it’s decreasing.

Match the intervals that describe the features of the piecewise function.

Learn how to analyze piecewise functions, including identifying intervals where the function is positive or negative, and where it is increasing or decreasing. This solution covers key concepts in piecewise functions, useful for students in Grades 9-12.

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