Lesson 1: Understand 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

Relations can be represented in three equivalent forms: ordered pairs (x, y), tables, and mapping diagrams with arrows.

To be a function, every input must have exactly one arrow pointing away from it (in a diagram) or correspond to exactly one output (in a table). It is completely acceptable for different inputs to share the same output.

Examples

• Valid Function: A table shows the months of the year (input) and the number of days in that month (output). Multiple months like January and March have the exact same output (31 days), which is perfectly allowed.

Property

When using a table to describe a function, the first variable (left column or top row) is the input, and the second is the output.
For the table to represent a function, every input value must correspond to exactly one output value.
It is acceptable for different inputs to have the same output, but a single input cannot have more than one output.

Examples

• A table lists student IDs and their corresponding GPAs. This is a function because each student ID (input) is linked to only one GPA (output).
• A table shows $x$ values of $\{2, 5, 2, 8\}$ and $y$ values of $\{3, 6, 4, 9\}$. This is not a function because the input $x=2$ corresponds to two different outputs, $y=3$ and $y=4$.
• A table shows month (input) and number of days (output). This is a function. Note that multiple months like January and March can have the same output (31 days), which is allowed.

Explanation

To check if a table shows a function, look for any repeated numbers in the input column. If you find a repeat, check if its output values are different. If they are, it's not a function. One input can't lead to two results.