Function: Definition

Definition

A function $f$ from a set $A$ to a set $B$ is a relation that assigns each element of set $A$ exactly one element of set $B$ .

This is written as:

$f: A \to B$

where:

$A$ is the domain
$B$ is the codomain
For every $a \in A$ , there exists one and only one $b \in B$ such that $f(a) = b$

Key conditions in the definition:

1. Every input must be assigned

— no element of the domain can be left unmapped.

2. Exactly one output for each input

— an input cannot map to more than one output.

3. The output must belong to the codomain

— the mapped value must be in the target set.

Example:

Let $A = \{1,2,3\}, \quad B = \{a,b,c,d\}$ If $f(1)=a,\quad f(2)=b,\quad f(3)=a$ then $f$ is a function because each element of $A$ has exactly one image in $B$ .

Non-example:

If $f(1)=a,\quad f(1)=b$ then this is not a function because the input $1$ has two outputs.

If an element of the domain does not map to anything, it is also not a function.

Main Content

1. Domain, Codomain, and Range

Domain

is the set of all allowed inputs for the function. It tells us where the function starts.

Codomain

is the set in which all outputs are expected to lie. It is the target set declared by the function.

Range

is the actual set of outputs produced by the function from the domain.

Example:

If $f: \{1,2,3\} \to \{1,2,3,4,5\}$ defined by $f(x)=x+1$ then:

Domain = $\{1,2,3\}$
Codomain = $\{1,2,3,4,5\}$
Range = $\{2,3,4\}$

This shows that range may be smaller than codomain.

Important distinction:

Domain tells us what inputs are accepted.
Codomain tells us the possible output universe.
Range tells us the outputs that actually occur.

2. Rule of Unique Output

A function must assign only one image to each element of the domain.
If one input has two or more outputs, the relation fails to be a function.
The uniqueness condition is what distinguishes functions from general relations.

Example of a valid function:

$f(x)=x^2$ For each real number $x$ , there is exactly one value of $x^2$ .

Example of an invalid function:

A relation such as $x \mapsto \pm \sqrt{x}$ is not a function if interpreted from $x$ to outputs, because a single input like $4$ gives two outputs: $2$ and $-2$ .

Set-based view:

If a relation $R \subseteq A \times B$ is a function, then for every $a \in A$ , there is exactly one pair $(a,b)$ in $R$ .

3. Representation of Functions

Functions can be represented in several ways: ordered pairs, tables, arrow diagrams, graphs, and formulas.
Each representation must still satisfy the definition of a function.
A clear representation helps identify whether the relation is valid and makes it easier to analyze function properties.

Ordered pairs:

$f=\{(1,a),(2,b),(3,a)\}$ This is a function because the first component of each pair is unique.

Arrow representation:

For $A=\{1,2,3\}$ and $B=\{x,y,z\}$ :

1 ───► x
2 ───► y
3 ───► x

This shows that each element of $A$ has exactly one arrow leaving it.

Graphical idea:

A function graph in Cartesian coordinates passes the vertical line test:

If any vertical line intersects the graph at more than one point, it is not a function.

Example of a function graph:

$y=x^2$

Example of a non-function graph:

$x=y^2$ when viewed as $y$ in terms of $x$ , because some $x$ -values correspond to two $y$ -values.

Working / Process

1. Identify the sets

Determine the domain and codomain clearly.
Check what elements are allowed as inputs and outputs.
Example: If $f: A \to B$ , then every element of $A$ must be considered.

2. Check the mapping rule

Verify that every input in the domain is assigned an output.
Ensure that no input has more than one output.
Confirm that each output lies in the codomain.

3. Verify function properties

Confirm whether the relation is one-to-one, onto, both, or neither if required.
Use ordered pairs, arrow diagrams, tables, or graphs to test validity.
If any input is missing or repeated with different outputs, the relation is not a function.

Advantages / Applications

Functions provide a precise way to model mathematical and real-world relationships between quantities.
They are the foundation for advanced topics such as algebra, calculus, discrete mathematics, and computer programming.
Functions help in theorem proving by allowing structured reasoning about inputs, outputs, and mappings between sets.

Summary

A function assigns each element of a domain exactly one element of a codomain.
Domain, codomain, and range are the core parts of a function.
A relation becomes a function only when every input has one unique output.

Important terms to remember: domain, codomain, range, relation, mapping, image, preimage, ordered pair