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**. The initial set or starting set is also called **domain**; the final set or arrival set, meanwhile, it can be called **codomain**.

Therefore, given a **set A** and a **set B**, a **function** is the association that occurs when each element of the **set A** (the domain) is assigned a single element of the **set B** (the codomain).

The generic element of the domain is known as **independent variable**; to the generic element of the codomain, such as **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** It is made up **nine specialists**. The contest rules 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 instance of the contest, there are **two finalists**. With all these data, we can affirm that there is a **function** what can we call **“choice”**, 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** makes each of the judges (elements of the domain) correspond to a single participant of 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 given unit, we do nothing but execute a function that depends directly on the variable *radio*, since the area is proportional to the square of it (it is 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 shorter the time.

## Analysis and representations

The idea that each element of the first **set** corresponds to only one of the second applies 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 the constructions that derive 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 with lowercase followed by the name of an arbitrary object in parentheses and also in lowercase, which represents the element of the domain whose name we want to find. **picture** 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 appeal to this **algorithm** or to an equation that best suits the needs of each case, even to 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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