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Mathematics has developed algebraic and transcendental logic processes to express the dependence between two elements or a set of elements. We are talking about math functions. So, for instance, the duration of the journey of a train from one city to another depends on the speed: the magnitude duration, here, is a function of speed.
The first magnitude (duration) is called a dependent variable while the second one (speed) is the independent variable. However, within this simple scheme, there are different types of functions.
In mathematics, a function (f) is the relation between a set of elements X (domain) and another set Y (codomain), so that each part of the domain corresponds to a single element of the codomain. Thus, the function has three bodies: two non-empty sets (X and Y), and a rule that relates both sets.
A function aims to discover how to obtain y through x. Functions are represented with this symbol f(x), and they represent the unknown quantity that we have to decipher in each value that we give x. In this way, we can say that f(x)=x.
There are many different types of functions depending on the elements it contains the way it relates, and the way we represent it. Here you have the easiest way of classifying them and the essential functions that are divided into algebraic functions and transcendental functions.
They are those functions which are represented with 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). So, an algebraic function solves a polynomial equation whose coefficients are in turn polynomials.
Thus, an algebraic function is one whose variable is acquired 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 expression, we distinguish between the various types of mathematical functions:
A related function is one whose expression is a polynomial of degree 1 and is represented as f(x)=ax+b and by a line in a graph. a corresponds to the slope of the line, and it tells us about its inclination, while b represents the independent variable. A related function example:
g(x)=3x-2h(x)=2x-7
To express a related function from its algebraic expression, we look for two pairs that belong to the function graph. These points are represented in the cartesian plane, and they join in a line, which gives us the graphic representation of the related function.
A related function can be increasing, when as the value of x increases so does the value of y, or decreasing, when as the value of x increases, the value of y decreases. When the value of y does not change when varying the value of x, we refer to a constant function.
A linear function is also expressed with a polynomial of degree 1 but, in this case, it does not have an independent term. It is represented as f(x)=ax and with a line that goes through the origin of coordinates. In other words, a linear function is one in which the function corresponds to ax being a any number. For example:
g(x)=2x ó h(x)=4x
To draw a linear function, the image of any value of the variable other than zero is found, the point corresponding to that pair ordered in the plane is marked, the line through point 0.0 and the previous point is drawn. Unlike the related function, this line always passes through the origin of coordinates.
The number that multiplies the variable is called proportionality constant: in g(x)=2x it would be 2. When the proportionality constant is positive, the straight line increases faster. If it is negative, the lower the constant, the faster it decreases. That is why the constant of proportionality is the slope of the line.
In the quadratic function, a second-degree polynomial is expressed with a single variable, and it is represented with a parabola whose elements are the axis of symmetry, the vertex, and the branches. Thus, for instance, a quadratic function is:
F(x)=3x^{2}+2x-2
For the graph representation of a quadratic function, we establish a table with some values of the function. The first thing to do is look for the vertex of the graph, and then pairs of points equidistant from the vertex. Accuracy depends on the number of points. It is also necessary to indicate the points of cuts with the axes.
If the independent term of the function increases, the parabola goes 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 close.
This type of function expresses a third-degree polynomial. The coefficients are rational numbers in which in the following function (x)=ax^{3}+bx^{2}+cx+d=0 the value of a is different to 0. This is an example of a cubic function:
Y=f(x)=x^{3}
To represent a graph of a cubic function, the function is evaluated for some x values. Then a table of values is made for the x-variable and the y-variable, a Cartesian plane is created, and the points are located by joining them forming the graph. Their particularity is that they cut the X-axis in one, two or three according to the number of real roots, and they cut the Y-axis in (0,d) given that f(0)=d.
A rational function is the one that can be written as a quotient of two polynomials and contains a variable in the denominator. In a given function of p(x) and q(x), they are polynomials, and q(x) is different from 0. Thus, for example, this is a representation of a rational function:
f(x)=1/x
In a rational function, an excluded value is any value of x that makes the value of the function y not defined. Thus, these values should be excluded from the function. If we get the function y=2/x+3 is -3. So, when x=3, the value for 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 function in the example above, the axis x and y are asymptotes, so the function graph will not touch the asymptotes.
Also called irrational functions, they are those that have in their definition a radical, a root. The most simple ones that are usually set as an example are the square roots with a real number other than 0, together with an a and a b.
First, the domain of the definition of the function has to be determined. As it is a square root, it will be all the values of x that make the radicand higher than or equal to zero. Then we have to check whether 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 sketch the function that should give us a lateral oblique shape. If we add a number to the variable x, the representation moves upwards, if we subtract it moves to the left or right, and if we multiply, it stretches or compresses.
a transcendental function is what does not satisfy a polynomial equation, which contrasts with the algebraic functions. Among the transcendental functions we can find the elemental type and the superior ones: the difference is that the elemental ones can be resolved with loads of operations.
Among these two types, these are the most relevant transcendental functions:
In math, the exponential term refers to the kind of growth whose rate increases faster and faster. The exponential function is the one in which the independent variable is an exponent. For instance:
f(x)=3^{1}=3
Exponential functions are used to analyze contexts in which a phenomenon grows exponentially (let's say, for example, demography). In the mother function f(x)=a^{x}, a is the base, and x is the exponent. The exponent is the independent variable that changes with time.
The exponential equation is the one in which the unknown quantity appears as an exponent. To solve it, we just need to equal the base: the properties of the powers are applied to achieve the same elevated base in the two members of the equation to appear at different exponents.
Logarithmic functions are usually used in math operations, in natural science, or in social science to compress the scale of measurements of magnitudes whose growth, of high acceleration, prevents a visual representation or the systematization of the represented phenomenon.
The logarithmic function is opposite to the exponential one, that is why its features are different: it only exists for values of x that are positive, without including zero. In the point x=1 the function is canceled, since log,1=0 on any basis. The logarithmic function of the base is always =1, and it is also continuous: increasing for a>1 and decreasing for a<1.
When in an equation the unknown quantity appears as the basis of a logarithm, it is called a logarithmic equation. Its method of resolution is the same as in the standard equations. An example of an algorithmic equation:
log_{a }f(x)=log_{a}g(x)
Trigonometric functions extend the definition of the reasons of trigonometry to real and complex numbers. They are used, especially in science, such as nautical, astronomy, cartography, or physics. More specifically, trigonometric functions are the quotient between two sides of a right triangle.
There are six essential functions in trigonometry: the last four correspond to the first two:
Sine: the relation between the length of the opposite side and the length of the hypotenuse.
Cosine: the relation between the length of the adjacent side and the length of the hypotenuse.
Tangent: the relation between the length of the opposite side and the adjacent.
Cotangent: the relation between the length of the adjacent side and the opposite.
Secant: the relation between the length of the hypotenuse and the length of the adjacent side.
Cosecant: the relation between the length of the hypotenuse and the length of the opposite side.
To define the values of these functions between 0 and 2π, a unit circumference created at the coordinate origin of a Cartesian plane is used. Trigonometric functions cosine and sine are defined as abscissa (x) and coordinate (y) of a point P of coordinates, belonging to the circle.