The mathematical sciences have developed algebraic and transcendental logical processes to express the dependency between two elements or sets of elements . They are mathematical functions. Thus, for example, the duration of the trip of a train from one city to another depends on the speed: the duration magnitude, here, is a function of speed.
The first magnitude (duration) is called the dependent variable while the second (velocity) is the independent variable. However, within this simple scheme, various types of mathematical functions are distinguished.
- You can also read: The 7 types of triangle (according to angles and sides).
1. What are mathematical functions?
2. Types of algebraic mathematical functions.
3. Types of transcendental mathematical functions. What are mathematical functions?
In mathematics, a function (f) is the relationship between a set of elements X (domain) and another set Y (codomain) , so that each element of the domain corresponds to a single element of the codiminium. Thus, the function consists of three fields: two non-empty sets (X and Y), and a rule that relates both sets.
The goal of a function is to find out how to get y through x. The functions are represented by the symbol f(x) and represent the unknown that we must decipherat each value we give to x. In this way we can say that f(x)=x. 9 types of mathematical functions
There are several types of mathematical functions depending on the elements they contain , the way they are related and the way we represent them. The simplest classification and the one that contains the most essential types of functions is divided into algebraic functions and transcendental functions. 1. Algebraic functions
They are those functions whose representation is an algebraic operation. In algebra, a polynomial is made up of a finite sum of products between variables (undetermined or unknown values) and constants (fixed numbers or coefficients). Well, an algebraic function solvesa polynomial equation whose coefficients are themselves polynomials .
In this way, an algebraic function is one whose variable y is obtained by combining a finite number of times the variable x together with operations of addition, subtraction, multiplication, division, raising to powers and extraction of roots. According to their composition and their expression, we distinguish between the various types of mathematical functions:1 1.1 Affine
function An affine function is one whose expression is a polynomial of degree 1 and is represented as f(x)=ax+b and by a straight linein a graph. A corresponds to the slope of the line and reports its inclination, while b represents the independent variable. An example of an affine function is the following:
To represent an affine function from its algebraic expression, we look for two ordered pairs that belong to the graph of the function . These points are represented in the Cartesian plane and are joined by a straight line, which gives us the graphical representation of the affine function.
An affine function can be increasing, when as the value of x increases, the value of y also increases, or decreasing, when as the value of x increases, the value of x decreases. When the value of y remains unchanged by changing the value of x , we speak of a constant function. 1.2 Linear
function A linear function also has a polynomial of degree 1 as its expression but, in this case, it does not have an independent term. It is represented as f(x)=ax and by a straight line passing through the origin of coordinates . That is, a linear function is one in which the function corresponds to x being any number. For example:
To draw a linear function, find the image of any value of the variable that is not zero, mark the point that corresponds to that ordered pair in the plane, draw the line that passes through the point 0,0and for the previous point. Unlike the affine function, this line always passes through the origin of coordinates.
The number that multiplies the variable is called the proportionality ratio: in g(x)=2x it would be 2. When the proportionality ratio is positive, the line grows faster the higher the ratio. If it is negative, it falls faster the smaller the ratio. That is why the proportionality ratio is the slope of the line . 1.3 Quadratic
function In the quadratic function a polynomial of degree 2 with a single variable is expressed , and it is represented by a parabola whose elements are the axis of symmetry, the vertex and the branches. So, for example, a quadratic function is:
For the graphic representation of the quadratic function we establish a table with some values of the function. First, the vertex of the graph must be found, and then pairs of points equidistant from the vertex . The precision depends on the number of points. It is also necessary to indicate the intersection points with the axes. If the independent term of the function is increased , the parabola moves up, and if the coefficient of degree 2 is changed, the branches of the parabola are inverted. If this coefficient is increased in absolute value, the branches are closed. 1.4 Cubic function
Also called third degree equation because it expresses a polynomial of degree 3. In this function the coefficients are rational numbersin which in the following given function f(x)=ax3+bx2+cx+d=0 the value of a is different from 0. It is a cubic function:
To represent a graph of a cubic function the function is evaluated for some values of x. Then a table of values is made for the variable x and the variable y, a Cartesian plane is created and the points are located joining them to form the graph. Their peculiarity is that they cut the X axis in one, two or three depending on the number of real roots, and they cut the Y axis in (0,d) given that f(0)=d. 1.5 Rational
function A rational function is one that can be written as the quotient of two polynomials and contains one variablein the denominator. In a given function p(x) and q(x) are polynomials, and q(x) is different from 0. Thus, for example, we have as a representation of a rational function:
In a rational function an excluded value is any value of x that makes the function y undefined. Thus, these values must be excluded from the function . If we take the function y=2/x+3 it is -3. Therefore, when x=-3 the value y is not defined. The domain of this function is the set of all real numbers except -3.
In algebra, an asymptote is a line that approaches the graph of the function but never touches it. In the example function we have given, the x and y axes are asymptotes, so that the graph of the function will approach without touching the asymptotes. 6. Radical
function Also called irrational functions, they are those that contain within their definition a radical, a root. The simplest ones that are usually given as an example are square roots with a real number other than 0 together with a and b.
First you have to determine the domain of definition of the function, which, since it is a square root, will be all the values of x that make the radicand greater than or equal to zero . Then we have to see if the function is positive or negative , which depends on the sign of the root that we have chosen.
Commenting on the point (-b/a, 0) in the positive or negative part we will performa sketch of the function that should give us a lateral oblique shape. If we add a number to the variable x the representation moves up, if we subtract it moves to the left or right, if we multiply it stretches or compresses. 2. Transcendental
functions A transcendental function is that which does not satisfy an equation of polynomials; this is in contrast to algebraic functions. We can find, within the transcendental functions, those of an elementary type and those of a superior type : the radical difference is that those of an elementary type allow to be resolved by means of a finite number of operations.
Within these two types, these are the most relevant transcendental functions: 2.1 Exponentials
In mathematics, the term exponential refers to the type of growth whose rate increases faster and faster. The exponential function is one in which the independent variable is an exponent . For example:
Exponential functions, therefore, are used to analyze contexts in which a phenomenon grows exponentially (say, for example, demographics). In the mother equation f(x)=ax we have that the base is a , while x is the exponent . The exponent is the independent variable that changes over time.
The exponential equation is one in which the unknown appears as an exponent. To solve it, just set the base equal: the properties of powers are applied to ensure that the same base raised to different exponents appears in the two members of the equation. 2.2 Logarithmic Logarithmic
functions are normally used in mathematical operations, in natural sciences or social sciences to compress the scale of measurements of magnitudes whose growth, of great acceleration , prevents a visual representation or the systematization of the represented phenomenon.
The logarithmic function is inverse to the exponential function, and therefore its features are opposite: it only exists for positive values of x , not including zero. At the point x=1 the function vanishes, since log,1=0 in any base. The logarithmic function of the base is always =1, and it is also continuous: increasing for a>1 and decreasing for a