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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 and range of the graph and decide whether it represents a function, I'll analyze the provided image.

### Domain:
The domain consists of all the possible \(x\)-values that the graph covers. From the graph:
- The graph stretches horizontally from \(x = -6\) to \(x = 6\).
- There are no breaks or gaps, so all \(x\)-values between \( -6 \) and \( 6 \) are included.

**Domain in set-builder notation:**
\[
\text{Domain} = \{ x \mid -6 \leq x \leq 6 \}
\]

### Range:
The range consists of all the possible \(y\)-values that the graph covers. From the graph:
- The lowest point of the graph is at \(y = -4\).
- The highest point appears to be at \(y = 6\).
- The graph covers all values between these points without any gaps.

**Range in set-builder notation:**
\[
\text{Range} = \{ y \mid -4 \leq y \leq 6 \}
\]

### Is it a function?
A graph represents a function if, for every \(x\)-value, there is exactly one corresponding \(y\)-value. 
- This graph does not pass the vertical line test (for example, at \(x \approx -4\) and \(x \approx 0\), a vertical line would intersect the graph at multiple points), meaning it fails to meet the criteria of a function.

**Answer:**
No, this graph does not represent a function.

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Do you have any further questions or need additional details? Here are some related questions you might consider:

1. What is the vertical line test, and how does it determine whether a graph is a function?
2. How do you find the domain and range for more complex graphs?
3. What is the difference between set-builder notation and interval notation?
4. Can a graph have a domain or range that is not continuous? How does that affect the graph's classification?
5. What are the implications if a graph fails the vertical line test?

**Tip:** Always use the vertical line test to quickly check if a graph represents a function by observing if any vertical line crosses the graph more than once.

Explore how to determine the domain and range of a graph and assess whether it represents a function using the vertical line test. This analysis covers basic graph interpretation, set-builder notation for domain and range, and criteria for function classification. Suitable for students in Grades 7-9.

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