A linear function is made up of first degree variables.
The notion of function has various uses. On this occasion, we are going to focus on the mathematical function : the relationship established between two sets, through which each element of the first set is assigned only one element of the second set, or none.
With this clear, we can advance the idea of a linear function . This is the name given to the mathematical function composed of first degree variables . It should be noted that a variable is a magnitude that, within the framework of 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 , making it possible to develop mathematical models that represent this link.
The starting set or initial set is called the domain , while the arrival set or final set is called the codomain . The independent variables are part of the domain ; the dependent variables , of the codomain. When equal changes in an independent variable correspond to equal variations in the dependent variable, it is called a linear function.
An example
Y = X + 2 is an example of a linear function. Suppose that in the domain we have the values 2 , 5 and 7 . If the function indicates that Y is equal to X + 2 , in the codomain we will find the values 4 , 7 and 9 :
X + 2 = Y
2 + 2 = 4
5 + 2 = 7
7 + 2 = 9
By taking this linear function to a graph in Cartesian coordinates, we will find an increasing straight line : as the values of X grow, the values of Y grow proportionally.
Linear functions, equations, and other concepts often cause difficulties in mathematics classes.
The linear function in geometry and algebra
The concept of a linear function is found in the field of analytical geometry and elementary algebra . The first is a branch of mathematics that focuses on the study of figures and their various properties, such as their areas, angles of inclination, 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 abstract structures and the combination of its 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 relationship that assigns a unique value to each instance of the variable and is composed of a polynomial, a sum or subtraction of a finite number of terms. An example of a polynomial function is f(x) = ax + b , where ax and 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: 0 , where the constant function is located, which always produces lines parallel or horizontal to the x axis; 2 , with the quadratic function , which when graphed generates parabolas ; 3 , to which the cubic function belongs, which is graphed in the form of cubic curves.
Returning to the linear function equation f(x) = ax + b , we can say that a and b are real constants and x , a real variable . The constant a is used to determine the inclination that the line will have when graphed (its slope ), while b indicates the point at which the line and the y- axis intersect.