Type of Functions

Definition

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

It is written as:

$f: A \to B$

where:

$A$ is the domain
$B$ is the codomain
the set of actual outputs in $B$ is called the range or image

For every $a \in A$ , there exists only one $b \in B$ such that:

$f(a) = b$

If one input has more than one output, then the relation is not a function.

Main Content

1. One-to-One Function (Injective Function)

A function is called one-to-one or injective if different elements of the domain always map to different elements of the codomain.

Meaning

If:

$f(a_1) = f(a_2)$

then it must follow that:

$a_1 = a_2$

This means no two distinct inputs can have the same output.

Example

Let:

$f(x) = 2x + 3$

If:

$f(1) = 5$
$f(2) = 7$
$f(3) = 9$

each input gives a unique output, so this is an injective function.

Graphical Idea

A function is injective if it passes the horizontal line test: any horizontal line intersects the graph at most once.

Importance

Helps in finding inverse functions
Used when each object must have a unique identifier
Important in cryptography, coding, and database design

2. Onto Function (Surjective Function)

A function is called onto or surjective if every element of the codomain has at least one pre-image in the domain.

Meaning

For every $b \in B$ , there exists at least one $a \in A$ such that:

$f(a) = b$

So, the entire codomain is covered by the function.

Example

Let:

$f(x) = x^3$

If the codomain is the set of all real numbers, then every real number has a cube root, so every value is achieved by some input. Hence, this function is onto $\mathbb{R} \to \mathbb{R}$ .

Key Point

A function is onto if the range = codomain.

Importance

Ensures every target value is reached
Useful in modeling complete systems
Common in proofs involving existence of solutions

3. One-to-One and Onto Function (Bijective Function)

A function is called bijective if it is both one-to-one and onto.

Meaning

A bijective function pairs every element of the domain with exactly one unique element of the codomain, and every element of the codomain is matched.

So:

no two inputs share the same output
no output is left unmatched

Example

Let:

$f(x) = x + 4$

from $\mathbb{R} \to \mathbb{R}$

It is one-to-one because different inputs give different outputs
It is onto because every real number $y$ can be written as $y = x + 4$ for some real $x$

Thus, it is bijective.

Why it matters

A bijection always has an inverse function
It establishes a perfect pairing between two sets
Used in counting, permutations, and equivalence of sets

Diagram for understanding bijection

Set A                     Set B
 a1  ------------------>   b1
 a2  ------------------>   b2
 a3  ------------------>   b3

Each element in A maps to a unique element in B, and every element in B is used.

Working / Process

1. Identify the domain and codomain

First, determine the input set and the target set.
Example: If $f: A \to B$ , then all inputs come from $A$ , and outputs must lie in $B$ .

2. Check the nature of mapping

See whether distinct inputs produce distinct outputs.
If yes, the function is one-to-one.
Then check whether every element in the codomain is reached.
If yes, the function is onto.

3. Classify the function

If only one-to-one → Injective
If only onto → Surjective
If both one-to-one and onto → Bijective
If neither condition holds fully, it is simply a function without those special properties

Advantages / Applications

Helps in mathematical classification

: Different types of functions help us understand the structure and behavior of mappings clearly.

Useful in inverse functions

: Only bijective functions have true inverses, which is important in algebra and problem solving.

Applications in computer science and cryptography

: Functions are used in algorithms, hash mappings, data encoding, encryption, and database keys.

Supports theorem proving

: Function properties are frequently used in proofs involving injectivity, surjectivity, and equivalence relations.

Real-world modeling

: Functions describe relationships such as age to height, marks to grades, or input to machine output.

Summary

Functions are special relations where each input has exactly one output.
Types of functions include one-to-one, onto, and bijective.
These classifications help determine whether outputs are unique, complete, or perfectly matched.
They are fundamental in mathematics, computer science, and logical reasoning.