General Mathematics Lesson 1: Functions and Relations

Study Notes

Functions and Relations

A relation is a set of ordered pairs (x, y).
The domain consists of all x-components (independent variable).
The range consists of all y-components (dependent variable).

Ordered Pairs Example

Given ordered pairs {(1,3), (2,4), (5,6), (7,8)}:
- Domain: {1, 2, 5, 7}
- Range: {3, 4, 6, 8}
Given ordered pairs {(-2,1), (0,-1), (5,3), (0,7)}:
- Domain: {-2, 0, 5}
- Range: {1, -1, 3, 7}

Definition of a Function

A function establishes a relationship where each x-value corresponds to exactly one y-value.

Representations of Functions

Mapping (one-to-one):
- Each domain x maps to one unique range y.
- Example:
  - Domain (x): 1, 3, 5, 7
  - Range (y): 2, 4, 6, 8
Mapping (one-to-many):
- A single domain x can map to multiple y-values.
- Example:
  - Domain (x): 2, 5, 7
  - Range (y): 3, 4, 6, 8
Mapping (many-to-one):
- Multiple domain x-values can correspond to a single range y-value.
- Example:
  - Domain (x): 2, 3, 4, 6
  - Range (y): 5, 7, 8

Function Identification

To determine if a set is a function, a vertical line test can be applied on its graph:
- A relation is a function if any vertical line intersects the graph at only one point.

Function Examples

Example of a function: {(1,1), (2,2), (3,3), (4,4)}
Example of a non-function: {(1,0), (0,1), (-1,0), (0,-1)}

Tables of Values

Functions can be presented in tabular format.
Example table with values:
- x: 1, 2, 3, 4
- y: 20, 40, 60, 80

Summary

All functions are relations, but not all relations are functions.
Understanding the distinction between functions and relations is essential for further mathematical concepts.

Description

Explore the fundamentals of functions and relations in this quiz designed for General Mathematics. Learn how to define and differentiate between functions and relations while understanding their representations. Additionally, determine the domain and range of given functions through ordered pairs.