Functions and Relations

Introduction

• Functions mean relationships.
• Relationship usually involves two parties.
• Represented often using an arrow diagram.

Arrow Diagram Example

• Example: 1 → 2, 2 → 3, 3 → 4
• Relation: Add 1 (x + 1)
• Function Represented as: f(x) = x + 1
• Object (x) and image (f(x)) concepts
• Bridge Concept: Object crosses 'bridge' (function) to get an image.

Function Representation Methods

1. Arrow Diagram
   • Starting (object, x)
   • Ending (image, f(x))
2. Ordered Pairs
   • Written like coordinate pairs: (object, image)
   • Example: (1, 2), (2, 3), (3, 4)
3. Graphs
   • X-axis represents objects
   • Y-axis represents images

Solving Functions and Relations Problem

Example: f(x) = 3x + 2

1. Find the image of zero
   • Given: Object x = 0
   • Image: f(x) = 3(0) + 2 = 2
2. Find the object with image 6
   • Given: Image f(x) = 6
   • Solve: 6 = 3x + 2 → x = 4/3
3. Object that maps onto itself
   • Given: f(x) = x
   • Solve: x = 3x + 2 → x = -1

Example: g(x) = |2 - 5x|

1. Find the image of object -2
   • Given: Object x = -2
   • Image: g(x) = |2 - 5(-2)| = 12
2. Find the object with image 7
   • Given: Image g(x) = 7
   • Solve: 7 = |2 - 5x| → Two values x = -1 or x = 9/5

Summary

• Functions describe relationships between objects and images.
• Represented by arrow diagrams, ordered pairs, and graphs.
• Solving involves understanding the relationship (bridge) and substituting values.

Homework/Next Steps

• Revise concepts of object and image in functions.
• Practice more problems on identifying objects and images.

Conclusion

• Functions establish relationships between inputs and outputs.
• Remember key concepts: object (x), image (f(x)), and methods of representation.
• Part 2 will continue covering more advanced topics and examples.