Bijective Function (Bijection)
2026-02-28 06:06 Diff
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Last updated on October 30, 2025

The bijective function is also referred to as a bijection or one-to-one correspondence, depending on the context. One-to-one function (injective) and one-to-one (bijective) correspondence are entirely different concepts. In this article, we will learn more about a bijective function.

What is a Bijective Function?

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A bijective function creates a perfect one-to-one match between two sets, such as set A and set B. If a function is said to be bijective, then it should satisfy the following two properties:

Injectivity: Each element from set A must be connected with a unique element in set B. In simple terms, no two elements from set A can connect with the same element in set B.
 


Surjectivity: Every element in set B must be the image of at least one element in set A. This function should map every element of set B to at least one element in set A, meaning all elements of set B are included. 

When a function satisfies these two properties, such as injectivity and surjectivity, then it is called a bijective function. A bijective function establishes a perfect one-to-one correspondence between two sets. 
 

Differences between Injective, Surjective, and Bijective Functions

In mathematics, a function links each input to an output. The way inputs and outputs are paired can be different. Based on this pairing, functions are classified as injective, surjective, or bijective. Let’s look at the differences between these functions.

Property

Injective Function (One-to-One)

Surjective Function(Onto)

Bijective Function (One-to-One & Onto)

What does it mean?

Every input gives a different output.

Every output has at least one input.

Every input gives a different output, and every output is used.

Does the output repeat?

No, outputs are not repeated.

Yes, outputs can be repeated.

No outputs are repeated, and each output is used exactly once.

Are all outputs used?

Not always

Yes, all outputs are used.

Yes, all the outputs are used.

Is it both One-to-One and Onto?

No

No

Yes

Symbol

↣

↠

⤖

Does it have an inverse?

Not always

No, not always.

Yes, always

Example

f(x) = 5x + 5

f(x) = x3

f(x) = x

The following image shows the difference between surjective, injective, and bijective functions:

What are the Properties of Bijective Function?

The main properties of a bijective function are injective and surjective, but other than that, some more properties of bijective functions are:

  • Inverse Exists
  • Unique Inverse
  • Preservation of Composition

Inverse Exists: The Inverse of a bijective function exists because the function pairs each element of the domain with a unique element of the codomain. The inverse reverses this mapping, taking an element from the codomain back to its original element in the domain.

Unique Inverse: The inverse of a bijective function is always unique, meaning there is only one function that can reverse the mapping.

Preservation of Composition: If two functions are bijective, then their composition is also bijective. 

How to Identify a Bijective Function?

For identifying a bijective function, we use two main processes, which are:

  • Injectivity
  • Surjectivity

Step 1: Check for Injectivity


Imagine two boxes, Box A and Box B.


Box A has: {1, 2, 3}


Box B has: {a, b, c}


We can use a function f to connect items from Box A to Box B,


f(1) = a


f(2) = b


f(3) = c


Now, check whether each number from Box A is connected with a different letter in Box B.


1 goes to a


2 goes to b


3 goes to c


Here, no two numbers go to the same letter. So it is injective, that is, one-to-one. 

Step 2: Check for Surjectivity


Next, see every letter in Box B is being used.


a is used by 1


b is used by 2


c is used by 3

No letter is left out. Each element in Box B is connected to something in Box A. This means the function is surjective; every element in Box B is covered. If a function passes both injective(one-to-one) and surjective(onto), then the function is said to be bijective. 


How to Represent Bijective Function in Graph?

Let us consider a bijective function as f(x) = x. It is a linear function with a slope that is equal to 1. Let us see it clearly through a graph given below:
 

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Tips and Tricks to Master Bijective Function

​​​​​Mastering bijective functions helps you understand how functions can form perfect one-to-one correspondences between sets. The tips below will guide you in identifying, verifying, and visualizing bijective relationships effectively.

  • Understand the definition clearly – A bijective function is both one-to-one (injective) and onto (surjective); know what each term means.

  • Check injectivity first – Ensure every element in the domain maps to a unique element in the codomain.

  • Verify surjectivity – Confirm that every element in the codomain has a pre-image in the domain.

  • Use diagrams – Draw arrow diagrams to visualize how domain elements connect with codomain elements.

  • Check the domain and range carefully – Misidentifying these often leads to incorrect conclusions about bijectivity.

Common Mistakes and How to Avoid Them in Bijective Functions

When learning about bijective functions, students often make a few mistakes by mixing up definitions or skipping important checks. Let’s look at some of these common mistakes and the ways to avoid them. 
 

Real Life Applications of Bijective Function

In real life, a bijective function is useful in many areas like technology, coding, mathematics, and even everyday tasks. Given below are some of the real-life applications of a bijective function.

  • Cryptography: When we send a secret message using a code, a bijective function helps to convert each letter or number into a unique symbol or value. Since it is a bijective function, the original message can be perfectly reversed using the inverse function, without losing or mixing up any data.
  • Computer Programming: In programming, bijective functions are used when each memory address must store exactly one value, and each value must have a unique memory address. This one-to-one matching prevents duplication or loss of data.
  • Database Management: In schools, banks, or hospitals, each person is given a unique ID number. A bijective function links one person to one ID, and vice versa. 
  • Matching Games or Puzzles: In games like memory cards or matching puzzles, each card is paired with exactly one other. A bijective function can be used to design or solve these games by creating a one-to-one link between the cards.
  • Barcode Systems: Each product has a unique barcode, and the barcode scanner uses a function to link the code to the product. This function is bijective, so that no two products share the same code and every code identifies one product only. 
     

Download Worksheets

Problem 1

Is the function f(x) = 2x + 3, defined for real numbers, a bijective function?

Okay, lets begin

Yes, it is a bijective function.
 

Explanation

 Check Injectivity:
Suppose f(x1) = f(x2)
2x1 + 3 = 2x2 + 3
Subtract 3:  2x1 = 2x2 
Divide by 2: x1 = x2
So, the function is injective

Check Surjectivity:
Let y be any real number.
y = 2x + 3
x = y - 32, which is also a real number.
So, for every output y, there exists an input x. It is surjective.

Since it is both injective and surjective, it is a bijective function.
 

Well explained 👍

Problem 2

Is the function f(x) = x2 for x ∈ R a bijective function?

Okay, lets begin

 No, it is not a bijective function.
 

Explanation

Injective check:
Try f(2) = 4 and f(-2) = 4
Two different inputs give the same output. So, it is not injective.

Surjective Check:
The function produces only non-negative outputs, so negative numbers like -1 or -5 never appear as outputs. Because of this, the function is not surjective. 

Since it is neither injective nor surjective, it is not bijective.
 

Well explained 👍

Problem 3

Let set A = {1, 2, 3}, B = {a, b, c}. Function f is defined as: f(1) = a, f(2) = b, f(3) = c. Is the function bijective?

Okay, lets begin

Yes, this is bijective.
 

Explanation

 Every number from A goes to one unique letter in B. So it is injective. Every letter in B is also used, so it is surjective. 
Here, both the injective and surjective are true, so it is a bijective function.
 

Well explained 👍

Problem 4

If set A = {x, y, z}, set B = {1, 2, 3}, f(x) = 1, f(y) = 2, f(z) = 3. Is this function bijective?

Okay, lets begin

Yes, it is bijective
 

Explanation

Each input gives a different output. So it is injective. 
All outputs are used, so it is surjective. 
Since there is no repeat and nothing left out, the function is bijective. 
 

Well explained 👍

Problem 5

Set A = {1, 2} and set B = {a, b}. Function f is, f(1) = a, f(2) = b. Is this function bijective?

Okay, lets begin

 Yes, the function is bijective.
 

Explanation

 Each number goes to a different letter, and every letter is used. So the function is a bijective function because it is both injective and surjective. 
 

Well explained 👍

FAQs of Bijective Function

1.What is a bijective function?

It is a perfect match between the two sets. Every item in the first set corresponds to a unique item in the second set, and every item in the second set is used; nothing is repeated or left out. 
 

2.How do I know if a function is bijective?

To see if a function is bijective, check two things: injective and surjective. If both conditions are true, the function is bijective. 
 

3.Can a bijective function be reversed?

Yes, a bijective function always has an inverse. You can go backward from the output to find the input.
 

4.Is every bijective function also injective and surjective?

Yes, a bijective function is both injective and surjective at the same time.
 

5.Why are bijective functions important?

Bijective functions help us to switch between two sets perfectly, build inverse functions, work with coding, algebra, and real-life situations like matching people to ID cards.  
 

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.