Thank you for uploading the images. The question asks to identify graphs that represent one-to-one functions.

Definition of a One-to-One Function

A one-to-one function passes the Horizontal Line Test, meaning that no horizontal line intersects the graph more than once. This ensures that every $y$-value corresponds to exactly one $x$-value.

Steps to Analyze:

I will examine each graph and determine if it passes the Horizontal Line Test.

1. Top Left (Oval Shape):

   • This is not a one-to-one function because a horizontal line intersects it in two places.
   • Not one-to-one.

2. Top Right (Parabola):

   • The graph of a parabola fails the Horizontal Line Test because horizontal lines intersect it twice.
   • Not one-to-one.

3. Second Row Left (Wave-like Graph):

   • A horizontal line can intersect this graph multiple times in some regions.
   • Not one-to-one.

4. Second Row Right (Increasing Curve):

   • This graph passes the Horizontal Line Test because every horizontal line intersects it at most once.
   • One-to-one.

5. Bottom (Straight Line):

   • A straight, non-horizontal line always passes the Horizontal Line Test.
   • One-to-one.

Final Answer:

• The graphs that represent one-to-one functions are:
  Second Row Right (Increasing Curve) and Bottom (Straight Line).

Would you like a step-by-step explanation or have any further questions?

Related Questions:

1. What is the Vertical Line Test, and how does it differ from the Horizontal Line Test?
2. Can a function fail the Horizontal Line Test but still be considered injective under certain transformations?
3. How do you algebraically test if a function is one-to-one?
4. What are examples of real-world phenomena modeled by one-to-one functions?
5. Why does a parabola fail the Horizontal Line Test?

Tip:

When testing for one-to-one functions graphically, always visualize both the Vertical and Horizontal Line Tests to ensure clarity about the graph's behavior.