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To determine the domain and range of the given piecewise function from the graph, let's analyze the x-values and y-values where the function is defined.

### **Domain:**
The domain of a function consists of all the x-values for which the function is defined. Let's look at the graph:

1. The function starts at \( x = -4 \) and continues to \( x = -2 \).
2. There is a gap between \( x = -2 \) and \( x = 0 \).
3. The function is defined again from \( x = 0 \) to \( x = 3 \).

- From the graph:
  - The function exists from \( x = -4 \) to \( x = -2 \) (both endpoints are included, indicated by filled circles).
  - Then from \( x = 0 \) to \( x = 3 \) (both endpoints are included).

Thus, the **domain** is:
\[ [-4, -2] \cup [0, 3] \]

### **Range:**
The range of a function consists of all the y-values the function takes.

1. For the interval from \( x = -4 \) to \( x = -2 \):
   - The y-values range from \( y = 1 \) to \( y = -3 \) (since there is a continuous line from point \((-4, 1)\) to \((-2, -3)\)).
   
2. For the interval from \( x = 0 \) to \( x = 3 \):
   - The y-values range from \( y = -2 \) to \( y = 2 \) (from point \((0, -2)\) to \((3, 2)\)).

- Combining these intervals:
  - The range from \( y = -3 \) to \( y = 1 \) (for the first segment).
  - The range from \( y = -2 \) to \( y = 2 \) (for the second segment).

Thus, the **range** is:
\[ [-3, 1] \cup [-2, 2] \]

### **Final Answer:**
- **Domain:** \[ [-4, -2] \cup [0, 3] \]
- **Range:** \[ [-3, 1] \cup [-2, 2] \]

Would you like a more detailed explanation or have any other questions?

### Additional Questions to Explore:
1. What is a piecewise function, and how is it defined mathematically?
2. How do you determine if an endpoint is included or excluded from a function's domain or range?
3. Can a function have multiple disjointed intervals for its domain or range?
4. What are some real-world examples where piecewise functions are applied?
5. How does one determine the continuity of a piecewise function?

**Tip:** Always pay attention to the type of circles (filled or open) on a graph, as they indicate whether the endpoints are included or excluded.

Determine the domain and range of the given piecewise function from the graph.

Learn how to determine the domain and range of a piecewise function by analyzing its graph. This problem focuses on key concepts related to functions and their intervals, making it ideal for students in Grades 9-11.

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