https://file.aiquickdraw.com/mathbot/file/1733709069122ywto538z.jpg

### Let's analyze the problem from the image and address the questions step by step.

The graph represents a function \( f(x) \) and has several features to analyze:

---

### **(a) Find the domain and the range of \( f(x) \):**
1. **Domain**: The domain is the set of all possible \( x \)-values for which the function is defined. From the graph, the function extends horizontally from \( x = -4 \) to \( x = 4 \), so:
   \[
   \text{Domain: } [-4, 4]
   \]

2. **Range**: The range is the set of all \( y \)-values that the function attains. From the graph, the lowest point is \( y = -3 \) and the highest point is \( y = 3 \), so:
   \[
   \text{Range: } [-3, 3]
   \]

---

### **(b) Intervals where \( f(x) \) is increasing, decreasing, or constant:**
- **Increasing**: The function is increasing where the slope of the curve is positive (going up). From the graph:
  \[
  \text{Increasing: } (-4, -3), (-2, 2), (3, 4)
  \]
  
- **Decreasing**: The function is decreasing where the slope is negative (going down). From the graph:
  \[
  \text{Decreasing: } (-3, -2), (2, 3)
  \]

- **Constant**: The function is constant where it has a horizontal segment (unchanging). From the graph:
  \[
  \text{Constant: } \text{No intervals where \( f(x) \) is constant.}
  \]

---

### **(c) Local minima and maxima:**
1. **Local Maximum**: Points where the function transitions from increasing to decreasing:
   - At \( x = -2 \), \( f(-2) = 1 \).
   - At \( x = 4 \), \( f(4) = 3 \).
   \[
   \text{Local Maxima: } (-2, 1), (4, 3)
   \]

2. **Local Minimum**: Points where the function transitions from decreasing to increasing:
   - At \( x = -3 \), \( f(-3) = 0 \).
   - At \( x = 2 \), \( f(2) = -1 \).
   \[
   \text{Local Minima: } (-3, 0), (2, -1)
   \]

---

### **(d) Solve \( f(x) > 0 \):**
Find intervals where \( f(x) > 0 \) (above the \( x \)-axis):
\[
f(x) > 0 \text{ on: } (-4, -3) \cup (-2, 0) \cup (3, 4)
\]

---

### **(e) Solve \( f(x) < 0 \):**
Find intervals where \( f(x) < 0 \) (below the \( x \)-axis):
\[
f(x) < 0 \text{ on: } (-3, -2) \cup (0, 3)
\]

---

Let me know if you'd like to see any part explained in greater detail!  
### Related questions:
1. How do you determine critical points from a graph of \( f(x) \)?
2. How can you calculate the range algebraically for more complex functions?
3. What are the first derivative tests for identifying increasing and decreasing intervals?
4. What makes a point a local extremum, and how can you differentiate it from a global extremum?
5. How does solving \( f(x) > 0 \) relate to real-world applications?

### Tip:
When analyzing a graph, always carefully observe transitions at key points (local maxima, minima, and intercepts) to accurately describe behavior!

Analyze the graph of the function f(x) and solve for domain, range, intervals of increase and decrease, local extrema, and solve for f(x) > 0 and f(x) < 0.

Explore a detailed step-by-step solution for analyzing a graph of a function f(x). Learn how to determine the domain, range, increasing and decreasing intervals, local maxima and minima, and solve for inequalities f(x) > 0 and f(x) < 0. This is a comprehensive guide for students in Grades 9-12.

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