Lesson 1: Understanding Relations and Functions

Property

A relation is a set of ordered pairs. The set of the first components (inputs) is called the domain, and the set of the second components (outputs) is called the range.

A function is a special relation where each possible input value leads to exactly one output value.

Examples

• The relation {(1, 3), (2, 5), (3, 7)} is a function because each input (1, 2, 3) has exactly one output.
• The relation {(A, 1), (B, 2), (A, 3)} is not a function because the input 'A' is paired with two different outputs, 1 and 3.
• In a school, if each student's name is an input and their assigned homeroom number is the output, this is a function because each student is assigned to only one homeroom.

Property

In many real-world situations, there is a relationship between two quantities where one quantity (the input) determines the value of another quantity (the output). We can identify these input-output relationships by examining how changes in one variable correspond to changes in another variable.

Examples

Property

An input-output table organizes pairs of numbers that are related by a rule. The 'input' is the number you start with, and the 'output' is the result after applying the rule. We often use variables like $x$ for the input and $y$ for the output.