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106 LearnersLast updated on October 16, 2025
Injective Function
An injective function, or one-to-one function, is a function where each input is mapped to a unique output. In this article, we will learn about injective functions, their properties, graphs, and how to identify them.
What is Injective Function?
In mathematics, a function is a relationship between one input and one output. A function is said to be injective if every element in the domain maps to a unique element in the codomain, that is, no two different inputs have the same output. A function f: A → B is injective if, for all x1, x2 ∈ A, whenever f(x1) = f(x2), then x1 = x2.
Difference Between Injective and Surjective Functions
In mathematics, two important types of functions are injective and surjective. They differ in the way the elements of the domain are associated with the elements in the codomain. In this section, we will learn how injective and surjective functions differ.
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Injective Function |
Surjective Function |
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An injective function maps every element of the domain to a unique element in the codomain, where no two inputs share the same output. |
A surjective function is one in which every element of the codomain is mapped to at least one element from the domain. |
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It is represented as f: A ↣ B |
It is represented as f: A↠B |
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Every input is mapped to different outputs |
Multiple inputs are mapped to the same output |
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Here, not all elements in the codomain are mapped |
Every element in the codomain should be mapped |
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For example, f(x) = 2x (R → R) |
For example, f(x) = x2 (R → R+) is not surjective unless codomain is adjusted. |
Properties of Injective Function
Injective functions have specific properties that help students identify and analyze them. Some of the key properties are:
- In an injective function, each element of the domain maps to a unique element in the codomain, and no two different inputs have the same output.
- Injective functions are often strictly increasing or decreasing, as monotonic behavior ensures that each element in the input is mapped to a unique output. However, not all injective functions are monotonic.
- An injective function has no direct link to critical points, that is the point where the derivative is zero or undefined, within its domain.
- When a function is injective and also surjective, then the function is bijective.
- The composition of two injective functions is also injective.
Graph of Injective Function
The graph of an injective function is used to visually represent the function. If a function is injective, its graph will pass the horizontal line test, which means that no horizontal line intersects the graph more than once.
To check whether the graph is injective or not, we use the horizontal line test. In the horizontal line test, we will check how many times a horizontal line crosses the graph.
- If the horizontal line crosses the graph once, then the function is injective.
- If the horizontal line crosses the graph at multiple points, then it's not injective.
How to Identify an Injective Function?
To check whether the function is injective, we use an algebraic method or the horizontal line test. Now let’s learn them in detail.
Algebraic Method
In the algebraic method, we will check whether the function has different inputs for the same output. In other words, check if f(x1) = f(x2) and show that x1 = x2.
For example, for f(x) = 2x + 3
Assuming f(x1) = f(x2)
2x1 + 3 = 2x2 + 3
Then x1 = x2
So, the function is injective
Horizontal Line Test
To check whether the function is injective, we use the horizontal line test. To check, follow these steps:
Step 1: First, represent the function in a graph
Step 2: Draw a horizontal line across the graph and check how many times the graph intersects the graph. If the line touches the graph once, the function is an injective function. If it intersects more than once, the function is not injective.
Real-World Applications of Injective Functions
Injective functions are used to understand the uniqueness and accuracy in various fields. In this section, we will learn about how it is used in real life.
- In database systems, injective functions help ensure that each input such as students' IDs, employee IDs, social security numbers maps to a unique individual. This one-to-one mapping prevents duplication and maintains data integrity.
- In inventory systems, the injective function is used to assign a unique barcode and QR code to each product.
- In banks, an injective function in account numbers or credit card numbers so that each number is unique to each customer.
- In programming and memory allocation, we use injective functions to assign a unique memory address to each variable. To prevent memory conflicts and overwriting in software systems.
- In healthcare, an injective function is used to assign a unique record number to patients to track their data. It helps in identifying the patients and ensuring accurate treatment and record keeping.
Common Mistakes and How to Avoid Them in Injective Functions
Understanding and learning the injection function is an important concept in algebra, calculus, and advanced mathematics. However, students make errors when working with an injective function. Here are a few common mistakes and ways to avoid them in an injective function.
Mistake 1
Confusing injective with surjective function
Students mix up injective functions with surjective functions by assuming that the injective function must map to every element in the codomain. This is wrong because in the injective function, every element in the input is mapped to a distinct codomain.
Mistake 2
Assuming all functions are injective
Sometimes students assume that a function is injective without verifying, especially when the functions that appear simple, like f(x) = x2. So, to avoid this error, always verify whether the function is injective or not by using the algebraic method or the horizontal line test.
Mistake 3
Misapplying the definition of injective
Students sometimes misapply or forget the definition of an injective function. For example, students assume x1 = x2, then f(x1) = f(x2) is true for any function. To avoid this, always understand and use the correct definition, that is, if f(x1) = f(x2), then it must follow that x1 = x2.
Mistake 4
Confusing with domain and codomain restriction
Students confuse the domain and codomain of an injective function and assume that a function is injective without checking the domain and codomain restrictions. To avoid this, always check the domain clearly, and students should memorize the definition of the injective function.
Mistake 5
Errors while using the horizontal line test
When using the horizontal line test to check whether the function is injective, students make errors by misinterpreting the graph. So, students should be careful when using the graphical method and also should remember that if any horizontal line intersects the graph more than once, then the function is not injective.
Solved Examples on Injective Function
Problem 1
Determine whether the function f(x) = 2x + 3 is injective.
The function f(x) = 2x + 3 is injective
Explanation
Checking if f(x1) = f(x2) and show that x1 = x2, to verify whether the function is injective
Let f(x1) = f(x2)
Then, 2x1 + 3 = 2x2 + 3
2x1 = 2x2 + 3 -3
2x1 = 2x2
Dividing by 2: x1 = x2
Since, x1 = x2, the function is injective
Problem 2
Check whether the function k(x) = ex is injective.
Yes, the function is injective
Explanation
The value of ex increases for all real x
So, if k(x1) = k(x2)
ex1 = ex2
Thus, x1 = x2
So, the function k(x) is injective
Problem 3
Is the function g(x) = x2 - 4x + 4 injective?
No, the function g(x) is not injective
Explanation
Given, g(x) = x2 - 4x + 4
Simplifying g(x): g(x) = (x - 2)2
Finding the value of g(x) by substituting x with real numbers
g(1) = (1 - 2)2 = 1
g(2) = (2 - 2)2 = 0
g(3) = (3 - 2)2 = 1
Here, the values of g(1) and g(3) are the same, so the function g(x) is not injective.
Problem 4
Check whether the function h(x) = sin x is injective on the interval [0, π].
No, the function h(x) is not injective
Explanation
To check whether the function h(x) is injective on [0, π]
For example, sin(π/6) = ½
sin(5π/6) = ½
As, the value sin(π/6) and sin(5π/6) are same, so the function is not injective
Problem 5
Check if the function r(x) = |x| is injective
No, the function r(x) is not injective
Explanation
To determine if a function is injective, we check whether it maps distinct inputs to distinct outputs.
Here, r(x) = |x|
If x = 2, r(2) = |2| = 2
If x = -2, r(-2) = |-2| = 2
Since the output is the same for two distinct inputs, the function is not injective.
FAQs on Injective Function
1.What is an injective function?
2.What is a surjective function?
3.What is the difference between an injective and a surjective function?
4.What are the methods to check if a function is injective?
5.Give an example of an injective function
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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.
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: He loves to play the quiz with kids through algebra to make kids love it.
