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given the following graph of a function, determine the domain and range.  Express your answer using interval notation.

To determine the domain and range of the function shown in the graph, let's analyze the key features:

### Domain:
The domain is the set of all possible x-values (input values) for which the function is defined.

- The graph shows that the function extends from the left side at \( x = -\infty \) (approaching negative infinity).
- The graph stops at \( x = 2 \), and there is an open circle at \( x = 2 \), which indicates that the function is not defined at \( x = 2 \).

Thus, the domain of the function is \( (-\infty, 2) \).

### Range:
The range is the set of all possible y-values (output values) for which the function is defined.

- The graph covers values of y from positive values near 4 and goes downward.
- The lowest point on the graph occurs at \( y = -5 \) with a solid point at \( (3, -5) \), indicating that \( y = -5 \) is included.
- The open circle at \( (2, -2) \) indicates that \( y = -2 \) is not included.

Thus, the range of the function is \( (-\infty, -2) \cup \{-5\} \).

### Final Answer:
- **Domain:** \( (-\infty, 2) \)
- **Range:** \( (-\infty, -2) \cup \{-5\} \)

Would you like more details or further explanation on any aspect of this problem?

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Here are some related questions for further practice:
1. How do you determine if a function is continuous based on its graph?
2. What is the significance of open and closed circles on a graph in terms of domain and range?
3. Can a function have the same y-value for different x-values in its domain? If so, give an example.
4. How does the concept of domain and range apply to real-world situations?
5. What changes would occur to the domain and range if the open circle at \( (2, -2) \) was filled in?

**Tip:** Always pay attention to open and closed circles on a graph; they indicate whether a point is included in the domain or range.

Given the following graph of a function, determine the domain and range. Express your answer using interval notation.

Learn how to determine the domain and range of a function based on its graph. This problem covers key mathematical concepts, including function behavior, open and closed circles, and interpreting graphs. Suitable for students in Grades 9-12.

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