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 able to make measurements and operate with the variables of the elements they study, in such a way that besides a discipline in itself it supposes next to the logic one of the bases of the scientific knowledge
But within mathematics very diverse processes and properties are studied, being between them the relation between two magnitudes or linked domains, in which a concrete result is obtained thanks to or in function of the value of a concrete element. It is about the existence of mathematical functions, which will not always 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.
- Related article: "14 mathematical riddles (and their solutions)"
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 short 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 . Said variables are symbolized from the last letters of the alphabet, X and Y, and respectively receive the name of domain and codomain.
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 classifications of functions that do not comply with this requirement).
Also, this function allows the creation of a representation in the form of a graphic 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 behavior of said variable.
As it happens when we say that something depends on or is based on something else (to give an example, if we consider that our grade in the math test is a 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 it.
In fact, the previous example is directly expressible in the form of a mathematical function (although in the real world the relationship is much more complex since in fact it depends on multiple factors and not only on the number of hours studied).
Main types of mathematical functions
Here we show some of the main types of mathematical functions, classified into different groups according to their behavior and the type of relationship established between the variables X and Y .
1. Algebraic functions
The algebraic functions are understood as 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 operations : addition subtraction, multiplication, division, potentiation or establishment (use of roots). Within this category we can find many types.
1.1. Explicit functions
Explicit functions are understood to be 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 relation in which the domain x influences .
1.2. Implicit functions
Unlike in 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 related.
1.3. Polynomial functions
Polynomial functions, sometimes understood as synonymous with algebraic functions and others as a subclass of these, integrate the set of types of mathematical functions in which To obtain the relationship between domain and codomain, it is necessary to perform several operations with polynomials of different degree.
Linear or first-grade 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 graphic representation is a line that has to cut the coordinate axis by some point. The only variation will be the slope of said 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 there is a direct 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 cutoff point of the abscissa and slope, y = mx + a).
The 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 in one of the axes. The graphic representation is established as a parabola, and 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 . That is, there is no real variation depending on the value of both: the codomain will always be a constant, there is no domain variable that can introduce changes. Simply, y = k.
- Maybe you're interested: "Dyscalculia: the difficulty when it comes to learning mathematics"
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 annul the denominator of the division, which would not allow to obtain a value y.
In this type of functions appear known limits as asymptotes , which would be precisely those values in which there would be no domain or codomain value (that is, when y and x are equal to 0). In these limits, the graphic representations tend to infinite, without ever touching said 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 is introduced into a radical or root (which does not have to be square, since it is possible that it is cubic or with another exponent).
To be able to solve it we must bear in mind 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 than or equal to zero.
1.6. Functions defined by pieces
This type of functions are those in which the value of y 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 will differ.
2. Transcendent functions
Transcendental functions are those mathematical representations of relationships between magnitudes that can not be obtained through algebraic operations, and for which it is necessary to perform a complex calculation process in order to obtain their relationship . It mainly includes 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 indicated by its name, 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, that is, there is an increasingly accelerated growth. 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. Log 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 basis. 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 base 1 is equal to zero). The growth of the function is decreasing as the value of x increases. In this case y = loga x
2.3. 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 before a determined value x.
Another classification
The set of mathematical function types explained above take into account that for each value of the domain corresponds a single value of the codomain (ie each value of x will cause a specific value of y). However, although this fact is usually considered basic and fundamental, it is certain that it is possible to find some types of mathematical functions in which there may be some divergence as far as correspondences between x and y are concerned . Specifically we can find the following types of functions.
1. Injective functions
The name of injective functions is that type of mathematical relationship between domain and codomain in which each of the values of the codomain is linked only to a value of the domain. That is, x will only be able to have a single value for a certain value, or it may have no value (that is, a specific value of x may not be related to y).
2. Surjective functions
The surjective functions are all those 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 they can be more. It does not have to be necessarily injective (to be able to associate several values of x to the same y).
3. Bijective functions
The type of function in which both injective and surjective properties are given is named as such. That is to say, there is a single value of x for each and , and all domain values correspond to one of the codomain.
4. Non-injective and non-surjective functions
These types of functions indicate that there are multiple values of the domain for a specific codomain (that is, different values of x are going to give us the same y) at the same time other values of y are not linked to any value of x.
Bibliographic references:
- Eves, H. (1990). Foundations and Fundamental Concepts of Mathematics (3 edition). Dover
- Hazewinkel, M. ed. (2000). Encyclopaedia of Mathematics. Kluwer Academic Publishers.
