In terms of mathematics, when talking about a function, one should mean a certain relation between a number of different inputs and a number of appropriate outputs with the particular property that every output is absolutely related to the only one input. In most of areas of mathematics in contemporary science functions are considered to be different kinds of the key objects of investigation. As a matter of fact, there is a number of various methods how to describe or present that or another function. Some of the known functions can be determined with the help of particular formula or relevant algorithm providing the explanation of completing the output for particular input. At the same time, other functions can be also determined with the help of a picture, which is usually called as the graph of certain function. Theres are also functions that can be determined with the help of a table providing relevant outputs for the chosen number of inputs. Anyway, studying of mathematical functions is a very broad and complicated issue. In this article, you will get familiar with different mathematical functions and learn how to deal with the even and odd functions.
Mathematics provides a specific explanation of the functions that are known as the even and odd functions. According to this explanation, the even and odd functions are any functions that can meet the following requirement: the are able to satisfy certain symmetry relations. This explanation is especially actual when it comes to the theory of Fourier series, as well as to the power series, which are both the areas of mathematical analysis. As for the mathematical analysis, this is a specific branch of mathematical science that is primarily dedicated to the theory of change, and to a number of different other theories. Among such theories, are the theory of differentiation, the theory of analytic functions, the theory of measure, integration and so on. As a rule, all these theories are researched through the prism of real and complex numbers, as well as real and complex functions. The branch of mathematical analysis covers a wide range of elementary concepts and various techniques and methodologies of analysis.
If to discuss the functions in terms of geometry, the most commonly used even functions are such as cos (x), sin (x), etc. On the other hand, if to speak about this issue in the light of algebraic science, we should mention a number of properties that these functions are characterized by. Among such properties, are the following:
The even and odd functions always have the products, which have also properties depending on different mix of the functions. Look what properties these products have:
Within your studying of mathematics, whether it is geometry or algebra, you will probably face many assignments requiring different operations with functions. In order to deal with such assignments and manage to find out the solution of that or another mathematical problem, you will have to be able to identify, whether the functions that are provided in the assignment are even, whether they are odd or not. You may cope with it by means of looking at the given equation itself, or by means of exploring the given graph of the function provided in the assignment. As a matter of fact, there are two methods of dealing with this problem, which leads to the same result, although are different from each other. The first method of recognizing the even and odd functions is algebraic method and the second is the method of graphing. They are both absolutely straightforward and are equally helpful in solving this kind of mathematical problem, but the relevance of using that or another method depends on the information that you are provided with. In addition, it depends on the level of your skills in the field of algebra or graphing. In any case the best way to succeed in completing this kind of assignment is practice. It is the most helpful, effective and useful thing. In order to recognize the even and odd functions, you need to follow the next steps:
When undertaking the steps provided above, you need to keep in mind that in an even function both the original function and the function that you get after replacement and solving should be absolutely the same; in an odd function, the main condition is that all the signs, whether they are pluses or minuses, need to be absolutely opposite (in other words, where there was plus, there has to be minus and vice versa). Remember that all these steps represent the algebraic method of determining the type of the function. Below, you will get familiar with another method, that deals with graphing.
The graphing method of determining the type of the function foresees that you create a graph on the basis of the original function that is primarily provided in the assignment. Detailed steps of using this method are listed below:
When undertaking the steps provided above, keep in mind the following information. The statement «symmetric in regard to the y-axis» means that the graph that is located on the left-hand side of the y-axis is the absolutely opposite reflection of the graph that is located on the right-hand side of the y-axis. As for the statement «symmetric in regard to the origin», it means that the graph that is located lower on the left-hand side of the whole graph is the absolutely opposite reflection of the function that is located on the top right-hand side of the whole graph. At the same time, both of the portions should reflect each other like in the mirror.
The term «graph theory» is widely used both in computer science and in mathematics. Generally speaking this is the study of graphs, their specificities, properties, functions, and other characteristics. Graphs are considered to be the mathematical structures that are widely used with the purpose of modeling pairwise relations between different objects. In this context, such mathematical structures can be made of nodes, vertices or points, each of which provides the graph with that or another characteristics. Any graph can be both directed and undirected, which depends on the distinction of the vertices that are connected to the edge. Apart from this, graphs represent the major aspect of study that discrete mathematics is dedicated to.
Different kinds of graphing methods are used for solving problems far beyond the field of mathematics. In computer science, graphs play significant role in representing the networks of communication, in structuring and organizing data and so on. For example, graph may be used for creating of the web-sites. The structure of many web-sites is very often represented with the help of a directed graph, where links from one page to another and the web-pages are represented by vertices and directed edges. The same approach and graphing method can also be used for solving various problems in biology, designing of computer chips and a wide range of other areas. For the last several decades the development of algorithms of using graphing methods is a widely discussed issue in the field of computer science.
In addition, the usage of graphing methods is very popular in linguistics, as long as the natural language is closely connected to the discrete structure. It is also used in the area of sociology, when dealing with researches, statistics, describing the relationships between people, etc.
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