Understanding the meaning of the even and odd functions
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:
- In case if a function that you are dealing with belongs to the type of the even and odd functions, it will be equal to zero, regardless of where it is determined.
- You are dealing with an odd function, in case if the absolute value of the given function is an even kind of function.
- The property concerning specifically even functions is characterized by the fact that the sum of two functions of this kind is even, and any constant multiple of this kind of function is also even.
- The unique property of specifically odd function is that the sum of two functions of this type is odd, and that any constant multiply of this kind of function is also odd.
- If you have two odd functions, the difference between them will be also odd.
- If you have two even functions, the difference between these two types of functions will be also even. The even and odd functions sum will not be odd and also it will not be even, with the exception of the condition that one of the functions is zero.
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:
- If you have two even functions, their product will be also an even function.
- If you are given two odd functions, their product will also be an even function, as in the previously described property.
- If you are looking for the product of the even and odd functions, you will get the odd function.
How to recognize the even and odd functions
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:
- Write down the original function that is provided in your mathematical assignment. Plug «-x» into the written function. Then you should look at the written original function very carefully and locate the «x» variable appropriately. Every time you find «x» in the given function, you need to remove it and put «-x» on its place.
- Once you have replaced «x» with the «-x», you need to solve the equation that you got as a result of this operation. Here, you need to evaluate the equation that you have. Use all your knowledge concerning standard multiplication and your knowledge of exponents in order to get rid of all the negative signs that are not necessary in the equation.
- Once you have got the result, you have to analyze them attentively. You need to look at your final function and at the given function that you started with and compare them.
- In case if the final function looks quite the same as the given original function, you can be sure that you are dealing with the even function.
- In case if the final function is different from the original primarily given function (for example, if it is absolutely opposite to the original function, or in case if it is negative from the given function), then the function you are dealing with is odd.
- If none of the provided above results is relevant in your case, you can be sure that the function you are working with is neither odd nor even.
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.
Identifying the even and odd functions with the help of graphs
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:
- First of all, you need to plot the original function on a graph. You may use a piece of graph paper, if appropriate, or you may sketch an empty graph. In order to graph the function, you need to plug in different values for x and look for the relevant results for y with the purpose of creating pairs of coordinate.
- The next step will be comparing the values of x and -x. Take a look at your graph carefully and figure out where you need to locate different x values for the given function. You will have to allocate the position of every value point of -x that corresponds to the value point of x.
- Now that you have the results, you need to analyze them attentively. If your function is symmetric in regard to the y-axis, it means that the function is even.
- In case if your final function is symmetric in regard to the original function, it means that you’re dealing with the odd function.
- In case if none of the above-provided conditions can be seen, it means that the function provided in the assignment is neither even, nor odd.
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.
Common facts about the graph theory
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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