Understanding One-to-One Functions

Key Concepts

• Function: A relation where every input (x-value or domain) has exactly one output (y-value or range).
• One-to-One Function: A function where each input has a unique output, meaning every y-value has exactly one corresponding x-value.

Process to Determine One-to-One Functions

1. Test for Functionality:
   • Ensure every x-value has exactly one y-value.
   • If not, the relation is not a function, thus cannot be one-to-one.
2. Test for One-to-One Functionality:
   • Ensure every y-value has exactly one corresponding x-value.

Example Analysis

First Table

• x-values: -4, -2, 1 (1 is listed once although it appears twice)
• y-values: 3, 5, 9 (3 appears twice)
• Mapping:
  • x = -4 maps to y = 3
  • x = -2 maps to y = 5
  • x = 1 maps to y = 9 & y = 3
• Analysis:
  • Not a function since x = 1 maps to two different y-values (3 and 9).
  • Not a one-to-one function.

Second Table

• x-values: -4, -2, 1, 3 (no repetition)
• y-values: -5, 2, 0 (2 appears twice)
• Mapping:
  • x = -4 maps to y = -5
  • x = -2 maps to y = 2
  • x = 1 maps to y = 0
  • x = 3 maps to y = 2
• Analysis:
  • It is a function since no x-value maps to more than one y-value.
  • Not a one-to-one function since y = 2 maps to two different x-values (-2 and 3).

Third Table

• x-values: -2, -1, 0, 1 (no repetition)
• y-values: 3, 4, 5, 8 (no repetition)
• Mapping:
  • x = -2 maps to y = 3
  • x = -1 maps to y = 4
  • x = 0 maps to y = 5
  • x = 1 maps to y = 8
• Analysis:
  • It is a function since each x-value maps to only one y-value.
  • It is a one-to-one function since each y-value maps to only one x-value.

Conclusion

• Shortcut Insight: If there is no repetition in either x-values or y-values, the relation is a one-to-one function.
• Mapping Benefits: Mapping helps visualize and understand the relationships, aiding in recognizing functions and one-to-one functions.