General Mathematics Lesson 1: Functions and Relations

Study Notes

Functions and Relations

Definitions
- 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}
Another set of ordered pairs: {(-2, 1), (0, -1), (5, 3), (0, 7)}
- Domain: {-2, 0, 5}
- Range: {1, -1, 3, 7}

Function Characteristics

A function maps each element of set x to exactly one element in set y.

Function Representations

Mapping (One-to-One)
- Each domain value corresponds to one unique range value.
Mapping (One-to-Many)
- A single domain value corresponds to multiple range values.
Mapping (Many-to-One)
- Multiple domain values correspond to a single range value.

Function Validation

Function Test: Ordered pairs
- {(1, 1), (2, 2), (3, 3), (4, 4)} is a function.
- {(1, 0), (0, 1), (-1, 0), (0, -1)} is not a function (0 cannot map to both 1 and -1).
Table of values validation
- For table with x: -1, 0, 1, 2 and y: 2, -2, 3, 3 - not a function due to the value 3 being repeated for different x-values.

Graphical Representation

Vertical Line Test:
- A relation is a function if a vertical line intersects the graph at only one point.
- If a vertical line intersects at more than one point, it is not a function.

Important Relationships

All functions are relations, but not all relations are functions.

Description

This quiz explores the fundamentals of functions and relations in mathematics. It covers definitions, different representations of functions, and the determination of domain and range. Test your understanding of ordered pairs and their significance in identifying domains and ranges.