One-to-One Functions: Testing with Graphs

Introduction

• Objective: Determine which graphs represent one-to-one functions.
• Steps:
  1. Check if the graph is a function.
  2. Test if it is a one-to-one function.

Testing for Functions: Vertical Line Test

• Vertical Line Test: A graphical test to determine if a graph represents a function.
  • Pass a vertical line across the graph.
  • If any vertical line intersects the graph at more than one point, it fails the test and is not a function.
• Importance: Only graphs that pass this test are considered functions.

Testing for One-to-One Functions: Horizontal Line Test

• Horizontal Line Test: Used to check if a function is one-to-one.
  • Pass a horizontal line across the graph.
  • If any horizontal line intersects the graph at more than one point, it fails the test and is not a one-to-one function.
• Procedure: Perform the vertical line test first; if it passes, perform the horizontal line test.

Example Analysis

1. First Graph:

   • Vertical Line Test: Passes (only intersects at one point).
   • Horizontal Line Test: Fails (intersects at multiple points).
   • Conclusion: Is a function but not one-to-one.

2. Second Graph:

   • Vertical Line Test: Fails (intersects at multiple points).
   • Conclusion: Not a function, thus not one-to-one.

3. Third Graph:

   • Vertical Line Test: Passes (intersects at one point).
   • Horizontal Line Test: Passes (does not intersect at more than one point).
   • Conclusion: Is both a function and one-to-one.

4. Fourth Graph:

   • Vertical Line Test: Fails (intersects at multiple points).
   • Conclusion: Not a function, thus not one-to-one.

Summary

• Among the four graphs examined, only the third represents a one-to-one function.
• Future examples will be examined in subsequent sessions.