# Surjective function

In the area of mathematics, a **function** is a **relationship** that exists between a given **set** X, also known as the **domain** name, and another set of **elements** Y, known as the **codomain** , in such a way that each element x of the domain corresponds to a single element f (x) of the codomain. The function, defined in simpler and simpler language is a common **logical process** that is expressed as ** “depends on”** . The functions can be classified depending on the

**relationship**between the

**elements**that the

**domain has**, the

**codomain**and

**image**. Mathematical functions in real life can refer to everyday situations, such as the cost of a phone call that depends on its duration, or the cost of sending an order that depends on its weight. A

**function**is a

**relationship**between

**two sets**in which each element of the first set corresponds to a single element of the second set. A

**surjective function**, also known as

**surjective**,

**epjective,**or

**surjective**function is a function in which each result value has at least one

**source value**.

## What is a surjective function?

We say that a function is surjective or surjective if all the **elements** that are in the image, AND have **anti-image** . In other words, if for any y of the **image Y** there exists at least one **element x** of the image such that f (x) = y.

- Definition of surjective function
- Properties of the surjective function
- Applications
- How to know if it is a surjective function
- Examples

## Definition of surjective function

A function is surjective, surjective or exhaustive, it occurs when the **codomain** and the **path** coincide. Formally, it can be defined as follows:

**∀y ****∈ Cod f ****∃ x ****∈ Dom f | f (x) = y**

This means, for any **element** y for the **codomain** there is another element x of the domain such that y is the image of x times f. In mathematics, a function is surjective if it is applied over the entire **codomain** , that is, when each element of “Y” is the image of at least one element of “X”. To make it a little easier to understand, we can say that each **set** or starting element has to cover the **elements** of the **codomain** or **arrival set** . Each value in a surjective function has to have its respective partner in the target set to be a surjective function.

## Properties of the surjective function

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

It has **domain** of the **function** which is the set of values that the independent variable can take, that is, those values for which the function is defined.

Regarding the image of the function, it has **growth** , which includes the **increasing** that occurs if when x increases, y is also increased; and **decreasing** , what happens if when increasing the value of x, the value of y. It also has a constant, if by varying x, y remains the same.

## Applications

In our daily life, the surjective function can and is applied in different **daily ****situations** . For example, it is used in **optimization** problems in the **statements** of **multiplicative structure** problems or of the traveling agent, where all the nodes that go from side 1 to side 2 have to be occupied. In the area of **finance** when to each **investment portfolio** corresponds to one or more investors.

Imagine, for example, that we are in a warehouse or a **supermarket** where there are many different **products** , each product found in the supermarket has a unique **barcode** and that only belongs to that product. The starting point here are the **products** and the arrival **sets** will be the barcodes. This means that for a product that is in the supermarket, there is only one barcode.

## How to know if it is a surjective function

We know that a function is surjective or surjective when each of the **elements** that exist within the **range** is the **image** of at least one **element** of the **domain** of said function. Another way to determine when a function is surjective is if every element of **the** final **set ****Y** has at least one element of **the** initial **set ****X** to which it corresponds. This means that a function is surjective if the path of the function is the final set Y.