What is a mathematical function?

We explain what a mathematical function is, how it can be expressed, its variables, the types that exist and other characteristics.

1. What is a mathematical function?

A mathematical function (also called simply function) is the relationship between one magnitude and another , when the value of the first depends on the second.

For example, if we say that the value of the temperature of the day depends on the time at which we consult it, we will be unknowingly establishing a function between both. Both magnitudes are variable , but they distinguish between:

• Dependent variable: It depends on the value of the other quantity. In the case of the example, it is the temperature.
• Independent variable: It is the one that defines the dependent variable. In the case of the example, it is time.

In this way, every mathematical function consists in the relationship between an element of a group A and another element of a group B, provided that they are linked uniquely and exclusively. Therefore, this function can be expressed in algebraic terms , using signs as follows:

f: A → B

a → f (a)

Where A represents the domain of the function ( f ), the set of starting elements, while B is the co- domain of the function, that is, the arrival set. By f (a) the relationship between an arbitrary object a belonging to domain A , and the only object of B that corresponds to it (its image ) is denoted .

These mathematical functions can also be represented as equations , using variables and arithmetic signs to express the relationship between the magnitudes. These equations, in turn, can be solved, clearing their unknowns, or be graphically plotted.

2. Types of mathematical functions

Mathematical functions can be classified according to the type of correspondence between the elements of domain A and those of B, thus having the following:

• Injective function . Any function will be injective if elements other than domain A correspond to elements other than B , that is, that no element of the domain corresponds to the same image of another.
• Surjection . Similarly, we will talk about an overjective (or subjective) function when each element of domain A corresponds to an image in B , even if it implies sharing images.
• Bijective function . It occurs when a function is injective and overjective at the same time, that is, when each element of A corresponds to a single element of B , and there are no images associated in the codominium, that is, there are no elements in B that do not correspond to one in.