The 13 types of mathematical functions (and their characteristics)

Mathematics is one of the most technical and objective scientific disciplines that exist. It is the main framework from which other branches of science are capable of making measurements and operating with the variables of the elements that they study, in such a way that in addition to a discipline in itself it supposes, together with logic, one of the bases of knowledge scientific.

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 based on the value of an element concrete. It is about the existence of mathematical functions, which are not always going to have the same way of affecting or relating to each other.

It is because of that we can talk about different types of mathematical functions, of which we will talk throughout this article.

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Functions in mathematics: what are they?

Before going on to establish the main types of mathematical functions that exist, it results from It is useful to make a short introduction in order to make it 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 quantities. These variables are symbolized from the last letters of the alphabet, X and Y, and are respectively given the domain and codomain names.

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

Also, this function allows the creation of a representation in the form of a graph which in turn allows the prediction of the behavior of one of the variables from the other, as well as possible limits of this relationship or changes in the behavior of said variable.

As it happens when we say that something depends on or is a function of another something (for example, if we consider that our mark in the mathematics exam is in function of the number of hours we study), when we talk about a mathematical function we are indicating that obtaining a certain value depends on the value of another linked to the.

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 just the number of hours studied).

Main types of mathematical functions

Here we show you some of the main types of mathematical functions, classified into different groups according to its behavior and the type of relationship established between the variables X and Y.

1. Algebraic functions

Algebraic functions are understood as the set of types of mathematical functions characterized by establishing a relationship whose components are either monomials or polynomials, and whose relationship is obtained through the performance of relatively simple mathematical operations: addition subtraction, multiplication, division, empowerment or radication (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 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 relationship influenced by the domain x.

1.2. Implicit functions

Contrary to the previous ones, in the implicit functions the relationship between domain and codomain is not established directly, being necessary to perform various transformations and mathematical operations in order to find the way in which x and y are relate.

1.3. Polynomial functions

Polynomial functions, sometimes understood as synonymous with algebraic functions and sometimes as a subclass of these, make up the set of types of mathematical functions in which to obtain the relationship between domain and codomain it is necessary to perform various operations with polynomials of varying degrees.

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 at some point. The only variation will be the slope of said line and the point where the axis intersects, always maintaining the same type of relationship.

Within them we can find the identity functions, in which an identification between domain and codomain is directly given 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 affine functions (in which we can find alterations in the cut-off point of the abscissa axis and the slope, y = mx + a).

Quadratic or second degree functions 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 graphical representation is established as a parabola, and mathematically it is 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. That is, there is no real variation based on the value of both: the codomain will always be based on a constant, and there is no domain variable that can introduce changes. Simply, y = k.

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1.4. Rational functions

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

In this type of functions, limits known as asymptotes appear, which would be precisely those values ​​in which there would not be a domain or codomain value (that is, when y or x are equal to 0). In these limits, the graphical representations tend to infinity, without ever touching said limits. An example of this type of function: y = √ ax

1.5. Irrational or radical functions

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

To be able to solve it It should be taken into account that the existence of this root imposes certain restrictions on us., for example the fact that the values ​​of x will always have to cause the result of the root to be positive and greater than or equal to zero.

1.6. Piecewise Defined Functions

These types of functions are those in which the value of and changes the behavior of the function, there are 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 it, which will be the value from which the behavior of the function differs.

2. Transcendent functions

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

2.1. Exponential functions

As its name indicates, exponential functions are the set of functions that establish a relationship between domain and codomain in which a growth relationship is established at an exponential level, that is, there is increasing growth accelerated. the value of x is the exponent, that is, 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 it will be necessary to raise the base used in order to obtain the concrete number. Thus, logarithmic functions are those in which we are using the number to be obtained with a specific base as the domain. It 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 base 1 is equal to zero). The growth of the function is less and less as the value of x increases. In this case y = loga x

2.3. Trigonometric functions

A type of function in which the numerical relationship is established between the different elements that make up a triangle or a geometric figure, 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 types of mathematical functions previously explained take into account that for each value of the domain corresponds to a single value of the codomain (that is, each value of x will cause a specific value of Y). However, and although this fact is usually considered basic and fundamental, the truth is that it is possible to find some types of mathematical functions in which there may be some divergence in terms of correspondence between x and y. Specifically we can find the following types of functions.

1. Injective functions

Injective functions are called that 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. That is, x will only be able to have a single value for a given y-value, or it may have no value (that is, a specific value of x may not have a relationship with y).

2. Surjective functions

Surjective functions are all those in which each and every one of the elements or values ​​of the codomain (y) is related to at least one of the domain (x), although they may be more. It does not necessarily have to be injective (since several values ​​of x can be associated with the same y).

3. Bijective functions

It is called as such the type of function in which both injective and surjective properties occur. Namely, there is a unique value of x for each y, and all values ​​in the domain correspond to one in the codomain.

4. Non-injective and non-surjective functions

These types of functions indicate that there are multiple domain values ​​for a specific codomain (i.e. different values ​​of x will give us the same y) as well as 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. (2000). Encyclopaedia of Mathematics. Kluwer Academic Publishers.