The notion of **function** it has various uses. On this occasion, we are going to focus on the **mathematical function**: the **relationship** that is established between two sets, through which each element of the first set is assigned only one element of the second set, or none at all.

With this in mind, we can advance the idea of **lineal funtion**. This is how the **mathematical function** made of **first degree variables**. It should be noted that a variable is a quantity that, within a certain set, can take on any of the possible values.

## Characteristics of the linear function

Linear functions are represented by a **straight line** in the **Cartesian plane**. It is important to keep in mind that what functions do, ultimately, is express a **relationship between variables**, being able to develop mathematical models that represent this link.

The starting set or initial set is called **domain**, while the arrival set or final set is called **codomain**. The **independent variables** are part of the **domain**; the **dependent variables**, of the codomain. When equal changes of an independent variable correspond to equal variations of the dependent variable, we speak of a linear function.

## An example

**Y = X + 2** it’s a **example** linear function. Suppose that in the domain we have the values **two**, **5** Y **7**. If the function indicates that **Y** is equal to **X + 2**, in the codomain we will find the values **4**, **7** Y **9**:

*X + 2 = Y2 + 2 = 45 + 2 = 77 + 2 = 9*

By bringing this linear function to a **graphic** in Cartesian coordinates, we will find a **growing straight line**: as the values of **X**, the values of **Y**.

## The linear function in geometry and algebra

The concept of a linear function is in the realm of **analytic geometry** and in the **elementary algebra**. The first is a branch of mathematics that focuses on the study of figures and their various properties, such as their areas, inclination angles, distances, intersections, volumes, and points of division, among many other characteristics. In short, we can say that it is a very deep vision of geometric figures to know all their **data** in detail.

On the other hand we have elementary algebra, where those fundamental concepts of algebra are found, the branch of mathematics that focuses on the **structures** abstracts and the combination of their elements according to certain rules. For arithmetic, only elementary operations between numbers take place, such as addition, subtraction, multiplication and division; algebra adds the symbols that denote numbers, the so-called *variables*, and in this way opens the doors to endless possibilities.

The linear function is itself a **polynomial function**, a relation that assigns a unique value to each instance of the variable and that is made up of a polynomial, an addition or subtraction of a finite number of terms. An example of a polynomial function is **f (x) = ax + b**, where **ax** Y **b** are the terms of the **polynomial**.

As mentioned in a previous paragraph, the linear function always gives straight lines on the Cartesian axes; more precisely, the lines are oblique, and this is the characteristic of first-degree polynomial functions. We have three more degrees: the **0**, where the **constant function**, which always produces lines parallel or horizontal to the x-axis; the **two**, with the **quadratic function**, which when graphing it generates **parables**; the **3**, to which the **cubic function**, which is graphed in the form of cubic curves.

Returning to the linear function equation **f (x) = ax + b**, we can say that **to** Y **b** are real constants and **x**, a **variable** real. The constant **to** It is used to determine the inclination that the line will have when it is graphed (its *earring*), while **b** indicates the point at which the line and the axis intersect **Y**.