The 13 types of mathematical functions (and their characteristics)
A summary of the classification of the types of functions used in mathematics.
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, so that in addition to being a discipline in itself it is, together with logic, one of the bases of scientific knowledge.
But within mathematics very diverse processes and properties are studied, being among them the relation between two magnitudes or domains linked to each other, in which a concrete result is obtained thanks to or depending on the value of a concrete element. This is the existence of mathematical functions, which will not always have the same way of affecting or relating to each other.
It is for this reason that we can talk about different types of mathematical functionsof which we are going to talk about in this article.
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 existing between two variables or quantities. These variables are symbolized by the last letters of the alphabet, X and Y, and are respectively called domain and codomain.
This relationship is expressed in such a way as to seek the existence of an equality between both components under analysis, and in general implies that for each of the values of X there is a single result of Y and vice versa (although there are classifications of functions that do not meet this requirement).
Also, this function allows the creation of a representation in the form of a graph, which in turn allows the 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 this variable.
Just as it happens when we say that something depends on or is a function of something else (for example, if we consider that our grade in the math exam is a function of the number of hours we study), when we speak of a mathematical function we are indicating that obtaining a given value depends on the value of another value linked to it.
In fact, the above example itself is directly expressible in the form of a mathematical function (although in the real world the relationship is much more complex as it actually depends on multiple factors and not only on the number of hours studied).
Main types of mathematical functions
Here are some of the main types of mathematical functions, classified in different groups according to their behavior and type of relationship according to their behavior 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 relation whose components are either monomials or polynomials, and whose relationship is obtained through the performance of relatively simple mathematical operationsThe following functions are used: addition, subtraction, multiplication, division, potentiation 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 x domain for the corresponding value. In other words, it is the function in which directly we find an equation between the value of and a mathematical relation which is influenced by the domain x.
1.2. Implicit functions
Unlike the previous ones, in 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 related.
1.3. Polynomial functions
Polynomial functions, sometimes understood as synonyms of algebraic functions and sometimes as a subclass of the latter, make up the set of types of mathematical functions in which to obtain the relation between domain and codomain it is necessary to perform various operations with polynomials of different degrees. of different degrees.
Linear or first-degree functions are probably the simplest 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 must cut the coordinate axis at some point. The only variation will be the slope of this line and the point at which it cuts the axis, always maintaining the same type of relationship.
Within them we can find the identity functions, in which there is a direct identification between domain and codomain. in such a way that both values are always the same (y=x), linear functions (in which we only observe a variation of the slope, y=mx) and affine functions (in which we can find alterations in the intersection point of the abscissa axis and the slope, y=mx+a).
The quadratic functions or second degree are those that introduce a polynomial in which a single variable has a nonlinear behavior over time (rather, in relation to the codomain). From a given 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 relation between domain and codomain. That is to say, there is no real variation depending on the value of both: the codomain will always be a function of a constant, there being 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 non-zero polynomials. In these functions the domain will include all the numbers except those that cancel the denominator of the division, which would not allow to obtain a value and.
In this type of functions there appear limits known as asymptoteswhich would be precisely those values in which there would be no domain or codomain value (i.e. when y or x are equal to 0). At such limits, the graphical representations tend to infinity, never touching these limits. An example of this type of function: y= √ ax
1.5. Irrational or radical functions
They receive the name of irrational functions the set of functions in which a rational function appears introduced inside a radical or root (that does not have to be square, since it is possible that it is cubic or with another exponent).
In order to solve it we must take into account that the existence of such a root imposes certain restrictions on usfor 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
This type of functions are those in which the value of y changes the behavior of the function, there being two intervals with 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 mathematical representations of relationships between quantities 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 a function. a complex process of calculation must be carried out in order to obtain their relationship.. It includes mainly those functions that require the use of derivatives, integrals, logarithms or that have a type of growth that is continuously increasing or decreasing.
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 relationship of exponential growth is established, i.e. there is an increasingly accelerated growth. the value of x is the exponent, i.e. 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. Thus the logarithmic functions are those in which we are using as domain the number to be obtained with a specific base. 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 smaller and smaller as the value of x increases. In this case y=log x
Trigonometric functions
A type of function that establishes the numerical relationship 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.
Another classification
The set of types of mathematical functions explained above take into account that for each value of the domain corresponds to a single value of the codomain (i.e. each value of x will cause a specific 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 it is possible to find types of mathematical functions in which there may be some divergence in terms of x and y correspondences. Specifically, we can find the following types of functions.
1. Injective functions
The name injective functions is given to that type of mathematical relationship between domain and codomain in which each of the values of the codomain is linked to only one value of the domain. That is, x can only have a single value for a given y-value, or it can have no value (i.e. a particular value of x can have no relation to y).
2. Surjective functions
The surjective functions are all those functions in which each and every one of the elements or values of the codomain (y) are related to at least one of the domain (x), although there may be more.although there may be more. It does not necessarily have to be injective (as several values of x can be associated to the same y).
3. Bijective functions
This is the name given to the type of function in which both injective and surjective properties are given. That is to say, there is only one value of x for each yand all the values of the domain correspond to one of the codomain.
4. Non-injective and non-surjective functions
This type of functions indicate that there are multiple values of the domain for a particular 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.
Bibliographical references:
- Eves, H. (1990). Foundations and Fundamental Concepts of Mathematics (3 edition). Dover.
- Hazewinkel, M. ed. (2000). Encyclopaedia of Mathematics. Kluwer Academic Publishers.
(Updated at Apr 13 / 2024)