Mathematics is one of the most technical and objective disciplines of science. It is the main framework from which other branches of science can measure and operate with the variables of the elements they study, so that in addition to a discipline in itself lies on the side of logic one of the bases of scientific knowledge.

But in mathematics very diverse processes and properties are studied, among which the relation between two magnitudes or domains related to each other, in which a concrete result is obtained thanks to or based on the value of a concrete element. It is about the existence of mathematical functions, which will not always have the same way of affecting each other or of relating to each other.

That is why **we can talk about different types of mathematical functions**, Which we will talk about throughout this article.

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## Functions in mathematics: what are they?

Before moving on to establishing the main types of mathematical functions, it is useful to give a little introduction in order to clarify what we are talking about when we talk about functions.

Mathematical functions are defined as **the mathematical expression of the relation between two variables or quantities**. These variables are symbolized from the last letters of the alphabet, X and Y, and receive the domain and codomain names respectively.

This relation is expressed in such a way that one seeks the existence of an equality between the two components analyzed, and generally this implies that for each of the values of X there exists a unique result of I and vice versa (although the classifications exist that do not meet this requirement).

Also, this function **allows the creation of a representation in the form of a graph** which in turn makes it possible to predict the behavior of one of the variables in relation to the other, as well as the possible limits of this relation or the changes in behavior of this variable.

As happens when we say that something depends on or is based on something else (to give an example, if we consider that our score on the math exam is based on the number of hours we study) , when we speak of a mathematical function we indicate that obtaining a certain value depends on the value of another which is linked to it.

In fact, the same example above is directly expressible as a mathematical function (although in the real world the relationship is much more complex because it actually depends on multiple factors and not just on the number of hours studied. ).

## Main types of mathematical functions

Below we show you some of the main types of math functions, categorized into different groups **depending on its behavior and the type of relationship that is established between the variables X and Y**.

### 1. Algebraic functions

Algebraic functions are understood as the set of types of mathematical functions characterized to establish a relation whose components are either monomials or polynomials, and **whose relation is obtained by performing relatively simple mathematical operations**: Subtraction by addition, multiplication, division, empowerment or rooting (use of roots). In this category we can find many typologies.

#### 1.1. explicit functions

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

#### 1.2. implicit functions

On the contrary than in the previous ones, in the implicit functions the relation between domain and codomain is not established in a direct way, and it is necessary to carry out various transformations and mathematical operations in order to find the path in which xiy is linked.

#### 1.3. Polynomial functions

Polynomial functions, sometimes understood as synonyms for algebraic functions and in others as a subclass thereof, constitute the set of types of mathematical functions in which **to obtain the relation between domain and codomain it is necessary to perform several operations with polynomials** of different 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, it simply has a simple relation in which a value of x will generate a value of y, and its graphical representation is a straight line that must intersect the coordinate axis at some point. The only variation was the slope of this line and the point where it intersects the axis, always keeping the same type of relationship.

In them we can find the identity functions, **in which an identification between domain and codomain is given directly** such that the two values are always the same (i = x), the linear functions (in which only one variation of the slope is observed, i = mx) and the associated functions (in which one can find alterations at point of intersection of the x-axis and the slope, i = mx + a).

Quadratic or quadratic functions are those which introduce a polynomial in which a single variable behaves nonlinearly over time (or rather, with respect to the codomain). From a specific limit, the function tends to infinity on one of the axes. The graphical representation is defined as a parabola and is expressed mathematically as i = 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 according to the value of the two: the codomain will always go according to a constant, there is no domain variable that can make modifications. Simply, i = k.

#### 1.4. rational functions

The set of functions in which the value of the function is established from a quotient between polynomials other than zero are called rational functions. In these functions the domain will include all the numbers except those which cancel the denominator of the division, which would not make it possible to obtain a value and.

**In this type of function, there are limits called asymptotes**, Which would be precisely the values in which there would be no domain or codomain value (i.e. when yox equals 0). At these limits, the graphic representations tend towards infinity, without ever touching these limits. An example of this type of function: i = √ x

#### 1.5. Irrational or radical functions

Irrational functions are the set of functions in which a rational function appears introduced in a radical or a root (which does not need to be square, because it is possible that it is cubic or with another exponent).

To be able to solve it **we must keep in mind that the existence of this root imposes certain restrictions on us**, Such as the fact that the values of x should always cause the result of the root to be positive and greater than or equal to zero.

#### 1.6. Functions defined by pieces

These types of functions are those in which the value of i changes the behavior of the function, there are two intervals with very different behavior depending on the value of the domain. There will be a value that is not part of it, that will be by what value the behavior of the function differs.

### 2. Transcendent functions

The transcendental functions are those mathematical representations of the relations between quantities which cannot be obtained by algebraic operations, and for which **it is necessary to perform a complex calculation process to obtain its relation**. It mainly includes functions which require the use of derivatives, integrals, logarithms or which have a type of growth which increases or decreases continuously.

#### 2.1. exponential functions

As its name suggests, exponential functions are the set of functions that establish a relationship between domain and codomain in which a growth relationship is established at the exponential level, i.e. there is a increasingly accelerated growth. the value of x is the exponent, i.e. the way in which **the value of the function varies and increases over time**. The simplest example: i = ax

#### 2.2. logarithmic functions

The logarithm of any number is that exponent that will be needed to raise the base used to get the specific number. Logarithmic functions are therefore those in which we use as domain the number to be obtained with a specific base. **It is the inverse and inverse case of the exponential function**.

The value of x must always be greater than zero and not equal to 1 (because any logarithm to base 1 is equal to zero). The growth of the function decreases as the value of x increases. In this case i = loga x

#### 2.3. 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 more precisely the relationships that exist between the angles of a figure. In these functions we find the calculation of sine, cosine, tangent, secant, cotangent and cosecant before a given value x.

## Another classification

The set of types of mathematical functions explained above takes into account the fact that for each value of the domain corresponds a single value of the codomain (that is to say that each value of x will cause a specific value of i). However, and although this fact is often considered fundamental and fundamental, the truth is that it is possible to find it. **type of mathematical functions in which there may be some discrepancy as to the correspondences between x and y refers to**. More precisely, we can find the following types of functions.

### 1. Injective functions

We call them injective functions this type of mathematical relationship between domain and codomain in which each of the values of the codomain is linked only to one value of the domain. In other words, x can have only one value for a given value i, or it can have no value (i.e. a particular value of x can have no relation to i) .

### 2. Surjective functions

Surjective functions are all those in which **each of the elements or values of domain (i) is linked to at least one domain (x)**, Although they may be more. It does not have to be necessarily injective (in order to be able to associate several values of x itself i).

### 3. Bijective functions

It is called as such the type of function in which both injective and surjective properties occur. In other words, that is to say **there is a unique value of x for each i**, And all the values of the domain correspond to a codomain.

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

These types of functions indicate that there are multiple domain values for a particular codomain (i.e. different values of x will give us the same i) at the same time that other values of i are linked. to no value of X.

#### Bibliographical references:

- Eves, H. (1990). Fundamentals and Fundamental Concepts of Mathematics (3rd Edition). Dover.
- Hazewinkel, M. ed. (2000). Encyclopedia of Mathematics. Kluwer University Publishers.