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

Let's analyze the graph to determine the domain, range, and whether it represents a function.

### 1. **Domain**:
   The domain of the graph is the set of all possible \(x\)-values that the graph covers.
   - From the graph, we can see the \(x\)-values range from \(-6\) to \(7\).
   - The graph has a filled circle at \(x = 5\), indicating that the value is included, and an open circle at \(x = -6\), indicating that the value is not included.
   
   **Domain** (in set-builder notation):
   \[
   \{ x \mid -6 < x \leq 7 \}
   \]

### 2. **Range**:
   The range of the graph is the set of all possible \(y\)-values that the graph takes.
   - The \(y\)-values range from \(-3\) to \(8\).
   - At \(y = 8\), there is a closed circle indicating it is included in the range, and at \(y = -3\), there is a closed circle as well.
   
   **Range** (in set-builder notation):
   \[
   \{ y \mid -3 \leq y \leq 8 \}
   \]

### 3. **Is the graph a function?**
   To determine if the graph is a function, we use the vertical line test. A graph represents a function if no vertical line intersects the graph at more than one point.
   - In this graph, at some \(x\)-values (like \(x = 3\)), the graph intersects at two different \(y\)-values. This means the graph is **not** a function.

**Answer**: No, the graph is not a function.

Would you like more details or have any questions?

Here are some related questions:

1. What is the difference between domain and range in a function?
2. How do you perform the vertical line test to determine if a graph is a function?
3. Can a graph have the same domain but different ranges?
4. How does the inclusion or exclusion of boundary points affect the domain and range?
5. How would the graph change if the open circle at \(x = -6\) was a closed circle?

**Tip**: When determining if a graph is a function, always remember that each input \(x\) should correspond to exactly one output \(y\).

Explore the domain, range, and functionality of graphs in mathematics. Learn how to determine the domain and range using set-builder notation and analyze if a graph represents a function using the vertical line test. This problem illustrates concepts crucial for understanding functions in mathematics.

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