General Mathematics Lesson 1: Functions and Relations

Questions and Answers

What defines the domain of a relation?

The set of all X-components of the ordered pairs. (correct)

The set of all Y-components of the ordered pairs.

The relationship between X and Y values.

The output values associated with each input value.

Which of the following represents a function?

{(5,1), (5,2), (6,1)}

{(1,2), (1,3), (2,4)}

{(2,2), (3,3), (4,4)} (correct)

{(0,0), (1,1), (1,2)}

What is the range of the relation {(1,3), (2,4), (5,6), (7,8)}?

{1, 2, 3, 4}

{3, 4, 6, 8} (correct)

{2, 4, 5, 7}

{3, 4, 6, 8} (correct)

What is the purpose of the vertical line test?

To determine if a relation has more than one output for an input. Signup and view all the answers

Which of the following scenarios does NOT represent a one-to-one function?

Multiple inputs can share the same output. Signup and view all the answers

Given the relation {(1,0), (0,1), (-1,0), (0,-1)}, is it a function?

No, because input value 0 has multiple outputs. Signup and view all the answers

Which of the following is NOT a representation of functions?

Random pairs Signup and view all the answers

Identify the domain of the relation {(-2,1), (0,-1), (5,3), (0,7)}.

{-2, 0, 5} Signup and view all the answers

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.