To determine if a graph represents a one-to-one function, we use the Horizontal Line Test. If any horizontal line drawn through the graph intersects it more than once, the function is not one-to-one. If every horizontal line touches the graph at most once, then the function is one-to-one.

Let's analyze each graph:

1. Graph 1 (Parabola):

   • A horizontal line will intersect the graph at two points.
   • Not one-to-one.

2. Graph 2 (Increasing curve):

   • A horizontal line will intersect the graph at only one point.
   • One-to-one.

3. Graph 3 (Sine-like wave):

   • A horizontal line will intersect the graph at multiple points.
   • Not one-to-one.

4. Graph 4 (Piecewise function with multiple values at certain points):

   • Since there are two points on the same vertical line with different y-values, it violates the one-to-one rule.
   • Not one-to-one.

5. Graph 5 (Linear function):

   • A horizontal line will intersect the graph at only one point.
   • One-to-one.

6. Graph 6 (Discrete points):

   • Since these are discrete points, no horizontal line will intersect more than one point.
   • One-to-one.

In summary:

• Graph 1: No
• Graph 2: Yes
• Graph 3: No
• Graph 4: No
• Graph 5: Yes
• Graph 6: Yes

Would you like further explanation or have any questions?

Here are some related questions:

1. How does the vertical line test differ from the horizontal line test?
2. Can a function be one-to-one but not continuous?
3. What does it mean for a function to be bijective?
4. How can we determine if a function is one-to-one algebraically?
5. What role do derivatives play in determining if a function is one-to-one?

Tip: When checking graphs for one-to-one functions, always imagine sweeping a horizontal line across the graph to ensure no two points are intersected simultaneously.