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

## What is a mathematical function?

A mathematical function (also called simply a function) **It 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 day's temperature depends on the time at which we consult it, we will be unknowingly establishing a function between the two things. Both magnitudes are variables, but they are distinguished between:

**Dependent variable**It is the one that depends on the value of the other magnitude. In the case of the example, it is the temperature.**Independent variable**It is what defines the dependent variable. In the case of the example it is the time.

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

**f: A → B**

**a → f(a)**

Where *TO* represents the domain of the function (*F*), the set of starting elements, while *b* is the codomain of the function, that is, the arrival set. By *fa)* denotes the relationship between an arbitrary object *to* belonging to the domain *TO*and the only object of *b* that corresponds to him (his *image*).

These mathematical functions **can also be represented as equations** resorting to variables and arithmetic signs to express the relationship between magnitudes. These equations, in turn, can be solved, solving their unknowns, or they can be graphed geometrically.

## Types of mathematical functions

Mathematical functions can be classified according to the type of correspondence that occurs 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 the domain*TO*correspond to elements other than the*b*that is, no element in the domain corresponds to the same image of another.**Surjective function.**Similarly, we will speak of a surjective (or subjective) function when each element of the domain*TO*corresponds to an image in the*b*even if it means sharing images.**Bijective function**It occurs when a function is injective and surjective at the same time, that is, when each element of*TO*corresponds to a single element of*b*and there are no unassociated images left in the codomain, that is, there are no elements in*b*that do not correspond to one in A.

Continue with: Analytical geometry

#### References

- “Mathematical function” on Wikipedia.
- “Function concept” in ICT Resources.
- “What is a mathematical function. Types of functions” in Ecuador.
- “Definition of mathematical function” in GeoGebra.
- “What is a mathematical function?” (video) in MClases.