https://file.aiquickdraw.com/mathbot/file/1726774619038kioa4ml9.jpg

From the uploaded image, we need to analyze Graph 31 for the following components:

(a) **Domain and Range of the function \( f(x) \):**
- **Domain:** The domain of a function from a graph is determined by the set of all possible \( x \)-values which the graph covers. For Graph 31, the function appears to cover all \( x \)-values from approximately \( x = -1 \) to \( x = 3 \), including both endpoints.
- **Range:** The range is determined by the set of all possible \( y \)-values that the function outputs. From the graph, the minimum \( y \)-value is about \( -0.5 \) and the maximum \( y \)-value is approximately \( 2 \).

(b) **Intervals on which \( f \) is increasing and decreasing:**
- **Increasing:** The function \( f \) is increasing on intervals where as \( x \) increases, \( y \) also increases. From the graph, \( f \) is increasing from \( x = -1 \) to \( x = 0 \) and again from \( x = 2 \) to \( x = 3 \).
- **Decreasing:** The function \( f \) is decreasing on intervals where as \( x \) increases, \( y \) decreases. From the graph, \( f \) is decreasing from \( x = 0 \) to \( x = 2 \).

**Summary:**
- Domain: \([-1, 3]\)
- Range: \([-0.5, 2]\)
- Increasing on: \([-1, 0]\) and \([2, 3]\)
- Decreasing on: \([0, 2]\)

Do you have any questions or need more details on this? Here are some follow-up topics you might find interesting:
1. What determines the end behavior of a polynomial from its graph?
2. How can one identify local maxima and minima from a graph?
3. What role do critical points play in understanding the behavior of functions?
4. How would the analysis change if the graph extended beyond the shown limits?
5. Can discontinuities affect the domain and range of a function?

Tip: Always check for any symmetry in the graph as it can provide insights into the function's behavior and potential algebraic form.

Analyze Graph 31 to determine the domain, range, and intervals of increasing and decreasing.

Learn how to analyze a graph to find the domain, range, and intervals where the function is increasing or decreasing. This exercise helps develop a deep understanding of function behavior and graph interpretation, suitable for students in Grades 10-12.

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