https://file.aiquickdraw.com/mathbot/file/1727024303520pcepw50z.jpg

Let's analyze the graph step-by-step to address the questions posed.

1. **Determine the largest open interval(s) where the function is:**
   - **(a) Increasing**: Look for sections of the graph where the function rises as the \(x\)-values increase. From the image, the function is never increasing because it either decreases or remains constant.
   - **(b) Decreasing**: The function is decreasing where the graph falls as \(x\)-values increase. It decreases from \(x = -\infty\) to \(x = -3\) and from \(x = 4\) to \(+\infty\). The intervals are:
     \[
     (-\infty, -3) \cup (4, \infty)
     \]
   - **(c) Constant**: The function is constant where the graph is a horizontal line. This happens between \(x = -3\) and \(x = 4\). The interval is:
     \[
     (-3, 4)
     \]

2. **Domain**:
   - The domain is the set of all possible \(x\)-values for which the function is defined. From the graph, the function is defined for all real numbers, so the domain is:
     \[
     (-\infty, \infty)
     \]

3. **Range**:
   - The range is the set of all possible \(y\)-values. From the graph, the function takes values between \(y = 1\) and \(y = 6\). So, the range is:
     \[
     [1, 6]
     \]

### Summary:
- **(a) Increasing interval**: None.
- **(b) Decreasing interval**: \( (-\infty, -3) \cup (4, \infty) \)
- **(c) Constant interval**: \( (-3, 4) \)
- **(d) Domain**: \( (-\infty, \infty) \)
- **(e) Range**: \( [1, 6] \)

Would you like more details or have any questions? Here are some related questions to further your understanding:

1. How would the intervals change if the graph had different slopes?
2. What conditions make a function increasing or decreasing?
3. How is the domain of a function different from its range?
4. Can a function be constant over more than one interval?
5. What does it mean for a function to have an open interval of increase or decrease?

**Tip**: Always check both the shape of the graph and how the \(y\)-values behave as the \(x\)-values change to determine increasing, decreasing, and constant intervals.

Determine the largest open interval(s) of the domain over which the function shown in the graph is (a) increasing, (b) decreasing, and (c) constant. Then give the domain and range.

Learn how to determine the largest open intervals of a function where it is increasing, decreasing, or constant using the graph. This problem also covers domain and range concepts, suitable for high school mathematics.

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