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

Would you like further details on any of these conclusions?

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.