https://file.aiquickdraw.com/mathbot/file/1730135214934x3g1y337.jpg

To determine if each graph is a one-to-one function, we apply the **Horizontal Line Test**: if any horizontal line intersects the graph more than once, it is *not* a one-to-one function.

### Analysis

1. **Graph 1**:
   - This appears to be a scatter plot of points, none of which have repeated \( y \)-values.
   - **Conclusion**: Yes, this graph represents a one-to-one function.

2. **Graph 2**:
   - This graph is a V-shaped function, likely an absolute value function. Horizontal lines will intersect the graph at two points in several places.
   - **Conclusion**: No, this graph is not one-to-one.

3. **Graph 3**:
   - This also appears to be a scatter plot of points. Some \( y \)-values are repeated (like at \( y = 2 \)), meaning it does not pass the horizontal line test.
   - **Conclusion**: No, this graph is not one-to-one.

4. **Graph 4**:
   - This looks like a continuous, smooth curve with no repeated \( y \)-values for each \( x \)-value.
   - **Conclusion**: Yes, this graph represents a one-to-one function.

5. **Graph 5**:
   - This graph appears to be a parabola opening downwards. Horizontal lines will intersect it twice, making it not one-to-one.
   - **Conclusion**: No, this graph is not one-to-one.

6. **Graph 6**:
   - This is also a parabolic shape, opening upwards. It fails the horizontal line test as well.
   - **Conclusion**: No, this graph is not one-to-one.

### Summary Table
| Graph | One-to-One? |
|-------|-------------|
| 1     | Yes         |
| 2     | No          |
| 3     | No          |
| 4     | Yes         |
| 5     | No          |
| 6     | No          |

Would you like further details on any of these conclusions?

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Here are some follow-up questions you might consider:
1. What is the horizontal line test, and why does it determine if a function is one-to-one?
2. Can you explain why a one-to-one function has an inverse that is also a function?
3. How would you identify a one-to-one function algebraically rather than graphically?
4. In what types of applications are one-to-one functions particularly useful?
5. How would the conclusions change if a graph was reflected across the x-axis or y-axis?

**Tip**: When verifying if a function is one-to-one, always check for any horizontal line intersecting more than once, as this would mean the function does not have unique outputs for each input.

For each function graphed below, state whether it is one-to-one.

Learn how to determine if a function is one-to-one by analyzing graphs using the Horizontal Line Test. This exercise covers graphical analysis and helps students understand the concept of one-to-one functions, ideal for Grades 9-12.

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