Mathematics is one of the most technical and objective scientific disciplines in existence. It is the main framework from which other branches of science are able to make measurements and operate with the variables of the elements they study, in such a way that in addition to being a discipline in itself, it is one of the bases of scientific knowledge.

But within mathematics, very diverse processes and properties are studied, among them the relationship between two magnitudes or domains linked to each other, in which a specific result is obtained thanks to or depending on the value of a specific element. This is the existence of mathematical functions, which will not always have the same way of affecting or relating to each other.

## Functions in mathematics: what are they?

Before going on to establish the main types of mathematical functions that exist, it is useful to make a small introduction in order to make clear what we are talking about when we talk about functions.

Mathematical functions are defined as the mathematical expression of the relationship between two variables or magnitudes . These variables are symbolised by the last letters of the alphabet, X and Y, and are given respectively the name of the domain and the co-domain.

This relationship is expressed in such a way that the existence of an equality between both analyzed components is sought, and in general it implies that for each of the values of X there is a single result of Y and vice versa (although there are function classifications that do not comply with this requirement).

Likewise, this function allows the creation of a representation in graphic form that in turn allows the prediction of the behaviour of one of the variables from the other, as well as possible limits of this relationship or changes in the behaviour of said variable.

Just as it happens when we say that something depends on or is a function of something else (to give an example, if we consider that our grade in the mathematics exam is a function of the number of hours we study), when we speak of a mathematical function we are indicating that the obtaining of a certain value depends on the value of another one linked to it.

In fact, the previous example itself is directly expressible in the form of a mathematical function (although in the real world the relationship is much more complex since it actually depends on multiple factors and not only on the number of hours studied).

## Main types of mathematical functions

Below are some of the main types of mathematical functions, classified into different groups according to their behaviour and the type of relationship established between the variables X and Y .

### 1. Algebraic functions

Algebraic functions are understood to be the set of types of mathematical functions characterized by establishing a relationship whose components are either monomials or polynomials, and whose relationship is obtained by performing relatively simple mathematical operations : addition, subtraction, multiplication, division, enhancement or rooting (use of roots). Within this category we can find numerous typologies.

#### 1.1. Explicit functions

Explicit functions are understood to be all those types of mathematical functions whose relationship can be obtained directly, simply by substituting the domain x with the corresponding value. In other words, it is the function in which we directly find an equalization between the value of and a mathematical relation influenced by the x-domain .

#### 1.2. Implicit functions

Contrary to the previous ones, in the implicit functions the relation between domain and codomain is not established in a direct way, being necessary to carry out diverse transformations and mathematical operations in order to find the way in which x and y are related.

#### 1.3. Polynomial functions

The polynomial functions, sometimes understood as synonyms of the algebraic ones and in others as a subclass of these, integrate the set of types of mathematical functions in which to obtain the relation between domain and codomain it is necessary to make diverse operations with polynomials of diverse degree.

Linear or first degree functions are probably the easiest type of function to solve and are among the first to be learned. In them there is simply a simple relationship in which a value of x will generate a value of y, and its graphical representation is a line that has to cut the coordinate axis by some point. The only variation will be the slope of this line and the point where it cuts the axis, always maintaining the same type of relationship.

Within them we can find the identity functions, in which directly there is an identification between domain and codomain in such a way that both values are always the same (y=x), the linear functions (in which we only observe a variation of the slope, y=mx) and the related functions (in which we can find alterations in the cut-off point of the abscissa axis and the slope, y=mx+a).

Quadratic functions or second degree are those that introduce a polynomial in which a single variable has a non-linear behavior over time (rather, in relation to the codomain). From a specific limit, the function tends to infinity on one of the axes. The graphic representation is established as a parabola, and is mathematically expressed as y=ax2+bx+c.

Constant functions are those in which a single real number is the determinant of the relationship between domain and codomain . In other words, there is no real variation in the value of both: the codomain will always be a function of a constant, and there is no domain variable that can introduce changes. Simply, y=k.

#### 1.4. Rational functions

Rational functions are the set of functions in which the value of the function is established from a quotient between polynomials other than zero. In these functions the domain will include all the numbers except those that cancel the division denominator, which would not allow to obtain a y value.

In this type of function there are limits known as asymptotes , which would be precisely those values in which there would not be a domain or codomain value (i.e. when y or x is equal to 0). In such limits, the graphic representations tend to infinity, without ever touching such limits. An example of this type of function: y= âˆš ax

#### 1.5. Irrational or radical functions

The set of functions in which a rational function is introduced within a radical or root (which does not have to be square, since it may be cubic or with another exponent) are called irrational functions.

To be able to solve it we will have to take into account that the existence of this root imposes certain restrictions , such as the fact that the values of x will always have to cause the result of the root to be positive and greater or equal to zero.

#### 1.6. Defined functions in pieces

This type of functions are those in which the value of and changes the behavior of the function, there being two intervals with a very different behavior based on the value of the domain. There will be a value that will not be part of this, which will be the value from which the behavior of the function differs.

### 2. Transcendent functions

Transcendent functions are those mathematical representations of relations between magnitudes that cannot be obtained through algebraic operations, and for which it is necessary to carry out a complex process of calculation in order to obtain their relation . It includes mainly those functions that require the use of derivatives, integrals, logarithms or that have a type of growth that is growing or decreasing continuously.

#### 2.1. Exponential functions

As their name indicates, the exponential functions are the set of functions that establish a relationship between domain and codomain in which a relationship of growth at an exponential level is established, that is to say that there is an increasingly accelerated growth. The value of x is the exponent, that is to say the way in which the value of the function varies and grows over time . The simplest example: y=ax

#### 2.2. Logarithmic functions

The logarithm of any number is that exponent which will be necessary to raise the base used in order to obtain the specific number. Therefore, the logarithmic functions are those in which we are using as a domain the number to be obtained with a specific base. This is the opposite and inverse case of the exponential function .

The value of x must always be greater than zero and different from 1 (since any logarithm with a base of 1 is equal to zero). The growth of the function decreases as the value of x increases. In this case y=log x

#### Trigonometric functions

A type of function in which the numerical relationship between the different elements that make up a triangle or a geometric figure is established, and specifically the relationships that exist between the angles of a figure. Within these functions we find the calculation of the sine, cosine, tangent, secant, cotangent and cosecant at a given x-value.

## Other classification

The set of mathematical function types explained above take into account that for each domain value there is only one codomain value (i.e. each value of x will cause a particular value of y). However, although this fact is usually considered basic and fundamental, it is possible to find some types of mathematical functions in which there may be some divergence in terms of correspondences between x and y . Specifically, we can find the following types of functions.

### 1. Injection functions

Injective functions are those types of mathematical relationships between domains and co-domains in which each of the values of the co-domain is linked to only one value in the domain. That is, x can only have one value for a given value y, or it can have no value at all (that is, a particular value of x can have no relationship to y).

### 2. Surjective functions

Surjective functions are all those in which each and every element or value of the codomain (y) is related to at least one of the domain (x) , although they can be more. It doesn’t necessarily have to be injectable (since several values of x can be associated with the same y).

### 3. Biyective functions

This is called the type of function in which both injectable and surjective properties are given. That is, there is a single value of x for each y , and all the values of the domain correspond to one of the codomain.

### 4. Non-injective and non-surjective functions

These types of functions indicate that there are multiple domain values for a given codomain (i.e. different values of x will give us the same y) while other values of y are not linked to any value of x.

#### Bibliographic references:

• Eves, H. (1990). Foundations and Fundamental Concepts of Mathematics (3rd edition). Dover.
• Hazewinkel, M. ed. Encyclopaedia of Mathematics. Kluwer Academic Publishers.