Relationships Can Be Described As Relations Or Functions
Mathematics often begins with a simple question: how are two quantities connected?
You might connect a student to their number of siblings, a time to a distance travelled, or a number to its square.
Any rule (or set of pairings) that links elements from one set to elements of another set is called a relation.
Definition
Relation
A set of ordered pairs $\{(x,y)\}$ that links elements from one set (inputs) to elements of another set (outputs).
A function is a special kind of relation with an extra restriction: each input must give exactly one output.
Definition
Function
A relation in which every input value in the domain is mapped to one and only one output value in the range.
You will meet these ideas in three linked representations:
Mapping diagrams (arrows from inputs to outputs)
Sets of ordered pairs $(x,y)$
Graphs on a coordinate plane
A Relation Is A Set Of Ordered Pairs With A Domain And Range
A relation can be described using sets.
Suppose there is a set $A$ of possible inputs and a set $B$ of possible outputs.
A relation is a collection of ordered pairs $(x,y)$ with $x\in A$ and $y\in B$.
In this sense, the relation is the "rule" that maps elements of set $A$ onto elements of set $B$.
Domain And Range Describe The Inputs And Outputs
Definition
Domain
The set of all input values $x$ used in a relation or function.
Definition
Range
The set of all possible output values (often $y$-values) that the function can produce from its set of input values (the domain).
If a relation is given as a list of ordered pairs, you can read the domain and range directly:
Domain: collect all the first coordinates (the $x$-values).
Range: collect all the second coordinates (the $y$-values).
Example
Consider the relation $R=\{(1,4),(2,4),(3,7),(5,7)\}$.
Domain: $\{1,2,3,5\}$
Range: $\{4,7\}$
Notice that the output 4 appears twice and the output 7 appears twice.
That is allowed in a relation.
Common Mistake
Students sometimes think the range must have the same number of elements as the domain. It does not.
Multiple inputs are allowed to share the same output in a relation, and in a function too.
Mapping Diagrams Show The Type Of Relationship
A mapping diagram uses arrows to show how elements of set $A$ (inputs) are mapped to elements of set $B$ (outputs).
Four common patterns are useful vocabulary.
Definition
One-to-one relation
A relation where each input is mapped to a unique output, and no two different inputs share the same output.
Definition
One-to-many relation
A relation where at least one input is mapped to more than one output.
Definition
Many-to-one relation
A relation where two or more different inputs are mapped to the same output.
Definition
Many-To-Many Relation
A relation where at least one input maps to multiple outputs and at least one output is shared by multiple inputs.
Note
A function can be one-to-one or many-to-one.
It cannot be one-to-many or many-to-many, because a function never allows a single input to have two different outputs.
Real-Life Examples Of Each Mapping Type
In everyday language, these mapping types show up naturally:
One-to-one: each person in a class $\rightarrow$ their student ID number (assuming IDs are unique).
Many-to-one: each student $\rightarrow$ their number of siblings (several students can have 2 siblings).
One-to-many: a parent $\rightarrow$ their children (one parent corresponds to several children).
Many-to-many: students $\rightarrow$ the sports they play (a student can play several sports, and each sport has many students).
Analogy
Think of a function like a vending machine button: when you press one button (input), exactly one snack comes out (output).
If one button could sometimes release two different snacks, the machine would not behave like a function.
A Function Is A Relation With Exactly One Output Per Input
The defining rule is simple but powerful:
For every $x$ in the domain, there exists one and only one $y$ in the range.
This is why it is correct to say:
All functions are relations (they are sets of ordered pairs).
Not all relations are functions (some relations assign two different outputs to the same input).
Function Notation Describes The Output From An Input
A function is often written as $f(x)=y$, read as "$f$ of $x$ is $y$".
$x$ is the input
$y$ is the output
$f$ is the name of the function
Another common notation is $f: x\mapsto y$, which emphasizes the mapping idea.
Tip
When you see $f(x)$, treat it as a single object meaning "the output when the input is $x$".
It is not multiplication of $f$ and $x$.
Testing Whether A Set Of Ordered Pairs Is A Function
If a relation is listed as ordered pairs, it is a function if no input value repeats with a different output.
Example
Is $\{(1,2),(2,5),(1,7)\}$ a function? No.
The input $x=1$ maps to both 2 and 7.
One input producing two outputs breaks the function rule.
Graphs Reveal Functions Visually
When a relation is drawn on a coordinate plane, the domain corresponds to the $x$-axis and the range corresponds to the $y$-axis (often written $y=f(x)$ for functions).
The Vertical Line Test Matches The Function Definition
A graph represents a function if and only if every vertical line intersects the graph at most once.
Why this works:
A vertical line fixes a single input value $x$.
If that vertical line hits the graph twice, then that same input has two different $y$-values, so the relation is not a function.
Exam technique
In tests, the vertical line test is often the fastest way to decide if a drawn relation is a function.
Do not confuse it with the horizontal line test, which is used to decide if a function is one-to-one (important for inverses).
Example
The relation $y=4x-9$ is a function because for every $x$ there is exactly one value of $y$.
Its graph is a straight line, so any vertical line crosses it only once.
It is also a one-to-one function: different inputs give different outputs (the line is not horizontal).
Example
Pick two different inputs:
If $x=0$, then $y=-9$.
If $x=1$, then $y=-5$.
The outputs are different, which is consistent with one-to-one behavior.
Common Mistake
A straight line can fail to be one-to-one if it is horizontal (for example $y=3$), because many different $x$-values all give the same output.
That is still a function, but it is many-to-one.
Connecting The Representations Builds Stronger Understanding
The "function rule" looks different in each representation, but it is the same idea:
Mapping diagram: each input dot has exactly one arrow leaving it.
Ordered pairs: no $x$-value is paired with two different $y$-values.
Graph: passes the vertical line test.
Note
These tests are all equivalent because they all check the same condition: whether a single input can produce more than one output.
Active recall
For each relation, state whether it is a function and justify your answer.
$\{(2,1),(3,1),(4,1)\}$
$\{(0,5),(0,7),(2,9)\}$
For each equation, decide whether it defines $y$ as a function of $x$.
$y=x+2$
$x^2=y^2$ (Hint: solve for $y$)
Give one real-life example of a relation that is not a function.