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State the domain and range for ach graph and then tellif the graph is a function (wite yes or no). in set builder notation

To determine the domain, range, and whether the graph represents a function, let's analyze the provided graph:

### Domain
The domain of a function is the set of all possible input values (x-values). From the graph:

- The x-values start from -3 and go up to 3.
- The graph has endpoints at \(x = -3\) and \(x = 3\).

**Domain in Set-Builder Notation:**
\[
\{x \mid -3 \leq x \leq 3\}
\]

### Range
The range of a function is the set of all possible output values (y-values). From the graph:

- The y-values start from -4 and go up to 5.
- The graph has endpoints at \(y = -4\) and \(y = 5\).

**Range in Set-Builder Notation:**
\[
\{y \mid -4 \leq y \leq 5\}
\]

### Function Check
To determine if the graph is a function, we use the **Vertical Line Test**: if any vertical line drawn through the graph touches the graph at more than one point, it is **not** a function.

- In this graph, a vertical line drawn at \(x = 0\) intersects the graph at two points: \( (0, 0) \) and \( (0, -1) \).

Since a vertical line intersects the graph at more than one point, **this graph is not a function**.

### Summary
- **Domain:** \(\{x \mid -3 \leq x \leq 3\}\)
- **Range:** \(\{y \mid -4 \leq y \leq 5\}\)
- **Is it a function?** No

Would you like more details or have any questions?

Here are 5 related questions that expand on this information:
1. How do you determine if a graph is a function using the vertical line test?
2. What is the significance of open and closed dots on a graph regarding domain and range?
3. Can you explain how to convert set-builder notation into interval notation?
4. What are some examples of graphs that do not represent functions?
5. How does the range change if the graph shifts vertically up or down?

**Tip:** When analyzing a graph, always check the endpoints carefully to determine if they are included (closed dot) or excluded (open dot) in the domain and range.

Explore how to determine the domain, range, and function status of graphs using set-builder notation. This detailed analysis covers key concepts like the domain as the set of all possible input values, range as the set of all possible output values, and the vertical line test for function verification. Suitable for high school students and beyond.

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