Functions are very important in the field of mathematics.
A mathematical function is a relationship established between two sets , through which each element of the first set is assigned a single element of the second set or none . 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 the 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.
Examples of mathematical functions
Take the case of a talent contest whose jury is made up of nine specialists . The rules of the contest establish that each member of the jury must choose one participant as the winner, without the possibility of voting blank or choosing more than one. In the final stage of the contest, there are two finalists . With all this 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, in this way, is made up of nine elements (each of the judges), while the final set or co-domain presents two elements (the finalists). The “election” function means that each of the judges (domain elements) is assigned a single contest participant (codomain elements).
In more scientific terms, when we calculate the area of a circle, for example, which is the measurement of its surface expressed in a certain unit, we do nothing more 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 its speed; Note that in this case the proportion is inverse, since the higher the speed, the less 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. that derive from them. If we think about integers, for example, where the natural numbers go from 1 to the most infinite, in addition to 0 and the negative ones to the least infinite, we can affirm that only one square corresponds to each one of them, which is always a number. natural or zero: -3 squared is 9; 0 squared is 0; 7 squared is 49.
The mathematical function we are faced with in this case has on the one hand the set of integers and on the other hand the set of natural numbers. We usually 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 domain element of which we want to find its image in the co-domain. 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 expressed in the example of the talent show, it is a concept that we apply unconsciously in everyday life.