When we talk about the area of **mathematics** , especially in the field of **functions** , it is also very important to know that the **function** is the link that develops between two **sets** that are **different** , a link through which, to each of these elements from one set are assigned only one element from another set or none at all. The idea of **an injective function** , on the other hand, refers to the property that tells us that two **different elements** of a first set are concerned with two totally **different ****elements** of a **second.** set that is not equal to the first.

## What is the injective function?

The injective function is the type of function that indicates that the different **elements** that an initial set or domain has, correspond to **different** elements of the final set or **codomain** , and each of these do not have a pre-image of the **domain** .

- Definition of injective function
- Properties of the injective function
- Applications
- Check if it is an injective function
- Examples

## Definition of injective function

The injective function is also known as the **one-to-one** function . A function can become injective if each of the **elements** that the final set Y has has a single element of **the initial set** X to which it corresponds. This means in other words that there cannot be more than one value of X that has the same image Y. Not all the elements of **the final set** Y must correspond to some element that exists in the initial set X.

We can say that the injective function occurs when each of the elements that has the **domain** does not correspond or cannot have more than one image in the **codomain** . In the area of mathematics , a function f: X **⇒** Y is injective if elements that are different from the set X or **domain** , correspond to different elements in the set Y or **codomain** of f. This means that each of the elements of the set Y has at most one **pre-image** in X, or, what is the same, in the set X there cannot be two or more elements that have the **same image** .

## Properties of the injective function

Among the properties of the injective function we mention the following:

- The
**notion**of**correspondence**has a basic role in the concept of relationship and function. - If n is an
**odd**number , then its domain is the entire set of real numbers . - If n turns out to be an
**even**number , the domain will then be formed by the**values**that make the radicand positive or zero.

## Applications

Injective functions serve us or are applied in the correct **graphing** of the different **functions** ; if the function of a single **real** variable is injective, any **horizontal line** will intersect only at one point. They are also applied to know if the function is **invertible** . Also to be able to make a **classification** of **linear transformations** (injective monomorphism), epimorphism (surjective), isomorphism (bijective).

## Check if it is an injective function

In mathematics, a function is injective if given two points **x **_{a}** and x **

_{b}**:**

** f (x **

_{a}

**) = f (x**_{b}) ⇒ x_{a}= x_{b}For this reason we can say that the function is injective if it manages to fulfill the **values** of its domain **x **_{0}** ≠ x **

_{1}**⇒**

*f (x*

_{0}

**) ≠ f (x**

_{1}

**)****.**

To check if the function is injective, it can also be done by **graphically** checking the injectivity of the **function** , and this is done when any line that is parallel to **the X axis** cuts the same line, at most, in **a point** .

## Examples

Some examples of the injective function that we can observe in our daily life are the following:

- When the cables are connected to an amplifier, not all the existing holes will be connected to one of the cables, but some will, and there will never be two connections in the same hole.
- When determining user identifiers, each of these users will have a unique identifier and no two users can have the same id.

They are also injective functions:

- The
**area of a square:**f (side) = side^{2} - The
**length of**a**circumference**: f (radius) = 2 π radius - The
**area**of a**circle**: f (radius) = π radius^{2} - The
**cube function**: f (x) = x^{3}