Sin is an even or odd function. Even and odd functions

Even function.

Even A function whose sign does not change when the sign is changed is called x.

x equality f(–x) = f(x). Sign x does not affect sign y.

The graph of an even function is symmetrical about the coordinate axis (Fig. 1).

Even function examples:

y= cos x

y = x 2

y = –x 2

y = x 4

y = x 6

y = x 2 + x

Explanation:
Let's take a function y = x 2 or y = –x 2 .
For any value x the function is positive. Sign x does not affect sign y. The graph is symmetrical about the coordinate axis. This is an even function.

odd function.

odd is a function whose sign changes when the sign is changed x.

In other words, for any value x equality f(–x) = –f(x).

The graph of an odd function is symmetrical with respect to the origin (Fig. 2).

Examples of an odd function:

y= sin x

y = x 3

y = –x 3

Explanation:

Take the function y = - x 3 .
All values at it will have a minus sign. That is the sign x affects the sign y. If the independent variable is a positive number, then the function is positive; if the independent variable is a negative number, then the function is negative: f(–x) = –f(x).
The graph of the function is symmetrical about the origin. This is an odd function.

Properties of even and odd functions:

NOTE:

Not all features are even or odd. There are functions that are not subject to such gradation. For example, the root function at = √X does not apply to either even or odd functions (Fig. 3). When listing the properties of such functions, an appropriate description should be given: neither even nor odd.

Periodic functions.

As you know, periodicity is the repetition of certain processes at a certain interval. The functions describing these processes are called periodic functions. That is, these are functions in whose graphs there are elements that repeat at certain numerical intervals.

How to insert mathematical formulas on the site?

If you ever need to add one or two mathematical formulas to a web page, then the easiest way to do this is as described in the article: mathematical formulas are easily inserted into the site in the form of pictures that Wolfram Alpha automatically generates. In addition to simplicity, this universal method will help improve the visibility of the site in search engines. It has been working for a long time (and I think it will work forever), but it is morally outdated.

If, on the other hand, you constantly use mathematical formulas on your site, then I recommend that you use MathJax, a special JavaScript library that displays mathematical notation in web browsers using MathML, LaTeX, or ASCIIMathML markup.

There are two ways to start using MathJax: (1) using a simple code, you can quickly connect a MathJax script to your site, which will be automatically loaded from a remote server at the right time (list of servers); (2) upload the MathJax script from a remote server to your server and connect it to all pages of your site. The second method is more complicated and time consuming and will allow you to speed up the loading of your site's pages, and if the parent MathJax server becomes temporarily unavailable for some reason, this will not affect your own site in any way. Despite these advantages, I chose the first method, as it is simpler, faster and does not require technical skills. Follow my example, and within 5 minutes you will be able to use all the features of MathJax on your site.

You can connect the MathJax library script from a remote server using two code options taken from the main MathJax website or from the documentation page:

One of these code options needs to be copied and pasted into the code of your web page, preferably between the tags and or right after the tag . According to the first option, MathJax loads faster and slows down the page less. But the second option automatically tracks and loads the latest versions of MathJax. If you insert the first code, then it will need to be updated periodically. If you paste the second code, then the pages will load more slowly, but you will not need to constantly monitor MathJax updates.

The easiest way to connect MathJax is in Blogger or WordPress: in the site control panel, add a widget designed to insert third-party JavaScript code, copy the first or second version of the load code presented above into it, and place the widget closer to the beginning of the template (by the way, this is not at all necessary , since the MathJax script is loaded asynchronously). That's all. Now learn the MathML, LaTeX, and ASCIIMathML markup syntax and you're ready to embed math formulas into your web pages.

Any fractal is built according to a certain rule, which is consistently applied an unlimited number of times. Each such time is called an iteration.

The iterative algorithm for constructing a Menger sponge is quite simple: the original cube with side 1 is divided by planes parallel to its faces into 27 equal cubes. One central cube and 6 cubes adjacent to it along the faces are removed from it. It turns out a set consisting of 20 remaining smaller cubes. Doing the same with each of these cubes, we get a set consisting of 400 smaller cubes. Continuing this process indefinitely, we get the Menger sponge.

The dependence of the variable y on the variable x, in which each value of x corresponds to a single value of y is called a function. The notation is y=f(x). Each function has a number of basic properties, such as monotonicity, parity, periodicity, and others.

Consider the parity property in more detail.

A function y=f(x) is called even if it satisfies the following two conditions:

2. The value of the function at the point x belonging to the scope of the function must be equal to the value of the function at the point -x. That is, for any point x, from the domain of the function, the following equality f (x) \u003d f (-x) must be true.

Graph of an even function

If you build a graph of an even function, it will be symmetrical about the y-axis.

For example, the function y=x^2 is even. Let's check it out. The domain of definition is the entire numerical axis, which means that it is symmetrical about the point O.

Take an arbitrary x=3. f(x)=3^2=9.

f(-x)=(-3)^2=9. Therefore, f(x) = f(-x). Thus, both conditions are satisfied for us, which means that the function is even. Below is a graph of the function y=x^2.

The figure shows that the graph is symmetrical about the y-axis.

Graph of an odd function

A function y=f(x) is called odd if it satisfies the following two conditions:

1. The domain of the given function must be symmetrical with respect to the point O. That is, if some point a belongs to the domain of the function, then the corresponding point -a must also belong to the domain of the given function.

2. For any point x, from the domain of the function, the following equality f (x) \u003d -f (x) must be satisfied.

The graph of an odd function is symmetrical with respect to the point O - the origin. For example, the function y=x^3 is odd. Let's check it out. The domain of definition is the entire numerical axis, which means that it is symmetrical about the point O.

Take an arbitrary x=2. f(x)=2^3=8.

f(-x)=(-2)^3=-8. Therefore f(x) = -f(x). Thus, both conditions are satisfied for us, which means that the function is odd. Below is a graph of the function y=x^3.

The figure clearly shows that the odd function y=x^3 is symmetrical with respect to the origin.

Chart conversion.

Verbal description of the function.

Graphic way.

The graphical way of specifying a function is the most illustrative and is often used in engineering. In mathematical analysis, the graphical way of specifying functions is used as an illustration.

Function Graph f is the set of all points (x; y) of the coordinate plane, where y=f(x), and x “runs through” the entire domain of the given function.

A subset of the coordinate plane is a graph of some function if it has at most one common point with any line parallel to the Oy axis.

Example. Are the figures below graphs of functions?

The advantage of a graphic task is its clarity. You can immediately see how the function behaves, where it increases, where it decreases. From the graph, you can immediately find out some important characteristics of the function.

In general, analytical and graphical ways of defining a function go hand in hand. Working with the formula helps to build a graph. And the graph often suggests solutions that you won’t notice in the formula.

Almost any student knows the three ways to define a function that we have just covered.

Let's try to answer the question: "Are there other ways to define a function?"

There is such a way.

A function can be quite unambiguously defined in words.

For example, the function y=2x can be defined by the following verbal description: each real value of the argument x is assigned its doubled value. The rule is set, the function is set.

Moreover, it is possible to specify a function verbally, which is extremely difficult, if not impossible, to specify by a formula.

For example: each value of the natural argument x is associated with the sum of the digits that make up the value of x. For example, if x=3, then y=3. If x=257, then y=2+5+7=14. And so on. It is difficult to write this down in a formula. But the table is easy to make.

The method of verbal description is a rather rarely used method. But sometimes it happens.

If there is a law of one-to-one correspondence between x and y, then there is a function. What law, in what form it is expressed - by a formula, tablet, graph, words - does not change the essence of the matter.

Consider functions whose domains of definition are symmetrical with respect to the origin of coordinates, i.e. for anyone X out of scope number (- X) also belongs to the domain of definition. Among these functions are even and odd.

Definition. The function f is called even, if for any X out of its domain

Example. Consider the function

She is even. Let's check it out.

For anyone X the equalities

Thus, both conditions are satisfied for us, which means that the function is even. Below is a graph of this function.

Definition. The function f is called odd, if for any X out of its domain

Example. Consider the function

She is odd. Let's check it out.

The domain of definition is the entire numerical axis, which means that it is symmetrical about the point (0; 0).

For anyone X the equalities

Thus, both conditions are satisfied for us, which means that the function is odd. Below is a graph of this function.

The graphs shown in the first and third figures are symmetrical about the y-axis, and the graphs shown in the second and fourth figures are symmetrical about the origin.

Which of the functions whose graphs are shown in the figures are even, and which are odd?

Evenness and oddness of a function are one of its main properties, and evenness occupies an impressive part of the school course in mathematics. It largely determines the nature of the behavior of the function and greatly facilitates the construction of the corresponding graph.

Let us define the parity of the function. Generally speaking, the function under study is considered even if for opposite values ​​of the independent variable (x) located in its domain, the corresponding values ​​of y (function) are equal.

Let us give a more rigorous definition. Consider some function f (x), which is defined in the domain D. It will be even if for any point x located in the domain of definition:

• -x (opposite dot) also lies in the given scope,

• f(-x) = f(x).

From the above definition, the condition necessary for the domain of definition of such a function follows, namely, symmetry with respect to the point O, which is the origin of coordinates, since if some point b is contained in the domain of definition of an even function, then the corresponding point - b also lies in this domain. From the foregoing, therefore, the conclusion follows: an even function has a form that is symmetrical with respect to the ordinate axis (Oy).

How to determine the parity of a function in practice?

Let it be given using the formula h(x)=11^x+11^(-x). Following the algorithm that follows directly from the definition, we first of all study its domain of definition. Obviously, it is defined for all values ​​of the argument, that is, the first condition is satisfied.

The next step is to substitute the argument (x) with its opposite value (-x).
We get:
h(-x) = 11^(-x) + 11^x.
Since addition satisfies the commutative (displacement) law, it is obvious that h(-x) = h(x) and the given functional dependence is even.

Let's check the evenness of the function h(x)=11^x-11^(-x). Following the same algorithm, we get h(-x) = 11^(-x) -11^x. Taking out the minus, as a result, we have
h(-x)=-(11^x-11^(-x))=- h(x). Hence h(x) is odd.

By the way, it should be recalled that there are functions that cannot be classified according to these criteria, they are called neither even nor odd.

Even functions have a number of interesting properties:

• as a result of the addition of similar functions, an even one is obtained;
• as a result of subtracting such functions, an even one is obtained;
• even, also even;
• as a result of multiplying two such functions, an even one is obtained;
• as a result of multiplication of odd and even functions, an odd one is obtained;
• as a result of dividing the odd and even functions, an odd one is obtained;
• the derivative of such a function is odd;
• If we square an odd function, we get an even one.

The parity of a function can be used in solving equations.

To solve an equation like g(x) = 0, where the left side of the equation is an even function, it will be enough to find its solutions for non-negative values ​​of the variable. The obtained roots of the equation must be combined with opposite numbers. One of them is subject to verification.

The same is successfully used to solve non-standard problems with a parameter.

For example, is there any value for the parameter a that would make the equation 2x^6-x^4-ax^2=1 have three roots?

If we take into account that the variable enters the equation in even powers, then it is clear that replacing x with -x will not change the given equation. It follows that if a certain number is its root, then so is the opposite number. The conclusion is obvious: the roots of the equation, other than zero, are included in the set of its solutions in “pairs”.

It is clear that the number 0 itself is not, that is, the number of roots of such an equation can only be even and, naturally, for any value of the parameter it cannot have three roots.

But the number of roots of the equation 2^x+ 2^(-x)=ax^4+2x^2+2 can be odd, and for any value of the parameter. Indeed, it is easy to check that the set of roots of a given equation contains solutions in "pairs". Let's check if 0 is a root. When substituting it into the equation, we get 2=2. Thus, in addition to "paired" 0 is also a root, which proves their odd number.