# What is mathematical function?

## What Does mathematical function Mean

A mathematical function is a relationship that is established between two sets , through which each element of the first set is assigned a single element of the second set or none at all . The initial set or starting set is also called the domain ; the final set or arrival set, meanwhile, can be called a codomain .

Therefore, given a set A and a set B , a function is the association that occurs when each element of set A (the domain) is assigned a single element of set B (the codomain).
The generic element of the domain is known as an independent variable ; to the generic element of the codomain, as a dependent variable . This means that, within the framework of the mathematical function, the elements of the codomain depend on the elements of the domain.

Functions are very important in the field of mathematics.
Examples of mathematical functions
Take the case of a talent show whose jury is made up of nine specialists . The rules of the contest establish that each member of the jury must choose a participant as the winner, without the possibility of voting blank or choosing more than one. In the final instance of the contest, there are two finalists . With all these data, we can affirm that there is a function that we can call "election" , which assigns each member of the jury the finalist they select. The initial set or domain, thus, is made up of nine elements(each of the judges), while the final set or codomain presents two elements (the finalists). The "choice" function means that each of the judges (elements of the domain) corresponds to a single participant in the contest (elements of the codomain).
In more scientific terms, when we calculate the area of ​​a circle, for example, which is the measure of its surface expressed in a certain unit, we do nothing other than execute a function that depends directly on the variable radius , since the area is proportional to the square of this (obtained by multiplying it by pi ). Similarly, a car trip has a duration that depends on other variables , such as the speed of the car; note that in this case the proportion is inverse, since the faster the speed, the shorter the time.

Blackboard with equations and graphs.
Analysis and representations
The idea that each element of the first set corresponds to only one of the second is applied in the field of mathematical analysis, the branch of mathematics that focuses on the study of complex and real numbers, as well as their functions and constructions. derived from them. If we think of the whole numbers, for example, where the natural numbers from 1 to the most infinite enter, in addition to 0 and the negatives to the minus infinity, we can affirm that each of them corresponds only to one square, which is always a number natural or zero: -3 squared is 9; 0 squared is 0; 7 squared is 49.
The mathematical function before which we find ourselves in this case has, on the one hand, the set of whole numbers and on the other, the set of natural numbers. In general, we denote a function by indicating its name in lowercase followed by the name of an arbitrary object in parentheses and also in lowercase, which represents the element of the domain whose image we want to find in the codomain. If we return to the example from the previous paragraph, we could say that the function to find the square of a given integer is f (n) = n * n .
Therefore, to represent a function we can use this algorithm or an equation that best suits the needs of each case, even tables in which the values ​​of each set are grouped. We must not forget that the mathematical function is not something exclusive to the scientific field but, as is well expressed in the example of the talent show, it is a concept that we unconsciously apply in everyday life.

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