Mathematics Relations and Functions

Study Notes

Relations

A relation is a set of ordered pairs.
Ordered pairs are written as (input, output) where the input comes from a set called the domain, and the output comes from a set called the range.
The domain is the set of all possible input values.
The range is the set of all possible output values.
Relations can be represented in different ways:
- Using a set of ordered pairs.
- Using a mapping diagram.
- Using a graph.
- Using a table.
Example: {(1, 2), (2, 4), (3, 6)} is a relation where the domain is {1, 2, 3} and the range is {2, 4, 6}.

Functions

A function is a special type of relation where each input value (from the domain) corresponds to exactly one output value (in the range).
In other words, no two different ordered pairs can have the same first element.
Functions can be represented in the same ways as relations.
Vertical Line Test: A graph represents a function if any vertical line intersects the graph at most once.

Mapping Diagrams

Mapping diagrams visually represent relations and functions.
Arrows connect elements in the domain to elements in the range.
For a relation to be a function, each element in the domain must have exactly one arrow pointing to an element in the range.

Function Notation

Function notation is used to represent functions symbolically.
Common notation: f(x), where 'f' represents the function and 'x' represents the input value.
f(x) represents the output value of the function for the input value 'x'.

Types of Functions

Linear Functions: have a constant rate of change and can be represented by a straight line.
Quadratic Functions: have a squared term and can be represented by a parabola.
Cubic Functions: contain a cubed term.
Polynomials: generalize functions made up of sums and powers of x.
Exponential Functions: contain an exponent where x is the exponent.

Domain and Range of Functions

Domain: The set of all possible input values for which the function is defined.
Range: The set of all possible output values that the function can produce.

Identifying Functions

When given a table, consider if every input (x-value) produces only one output (y-value).
When examining a set of ordered pairs, ensure each x-value is associated with only one y-value.

Determining if a Relation is a Function

Use the vertical line test for graphs.
Check if each x-value in a set of ordered pairs or a table is paired with only one y-value.
Examine the mapping diagram; each element in the domain is assigned to only one element of the range.

Key Differences Between Relations and Functions

Relations allow multiple outputs (y-values) to correspond to a single input (x-value).
Functions ensure unique and single outputs(y-values) for each input (x-value).

Examples

Consider the relation {(1, 2), (2, 3), (1, 4)}. This is not a function because the input '1' is associated with two different outputs (2 and 4).
The relation {(1, 2), (2, 3), (3, 4)} is a function because each input is mapped to a unique output.

Applications of Functions and Relations

Real-world situations can be modeled by functions and relations.
For example, the relationship between distance traveled and time can often be described by a linear function.

Description

This quiz covers the concepts of relations and functions in mathematics. You'll learn about ordered pairs, domains, ranges, and how to identify functions. Test your understanding of representations like mapping diagrams, graphs, and tables.