Odd function. Function properties. General scheme for plotting functions

Function zeros
The zero of the function is the value X, at which the function becomes 0, that is, f(x)=0.

Zeros are the points of intersection of the graph of the function with the axis Oh.

Function parity
A function is called even if for any X from the domain of definition, the equality f(-x) = f(x)

An even function is symmetrical about the axis OU

Odd function
A function is called odd if for any X from the domain of definition, the equality f(-x) = -f(x) is satisfied.

An odd function is symmetrical with respect to the origin.
A function that is neither even nor odd is called a general function.

Function Increment
The function f(x) is called increasing if the larger value of the argument corresponds to the larger value of the function, i.e. x 2 >x 1 → f(x 2)> f(x 1)

Decreasing function
The function f(x) is called decreasing if the larger value of the argument corresponds to the smaller value of the function, i.e. x 2 >x 1 → f(x 2)
The intervals on which the function either only decreases or only increases are called intervals of monotony. The function f(x) has 3 intervals of monotonicity:
(-∞ x 1), (x 1 , x 2), (x 3 ; +∞)

Find intervals of monotonicity using the service Intervals of increasing and decreasing functions

Local maximum
Dot x 0 is called a local maximum point if for any X from a neighborhood of a point x 0 the following inequality holds: f(x 0) > f(x)

Local minimum
Dot x 0 is called a local minimum point if for any X from a neighborhood of a point x 0 the following inequality holds: f(x 0)< f(x).

Local maximum points and local minimum points are called local extremum points.

x 1 , x 2 - local extremum points.

Function Periodicity
The function f(x) is called periodic, with period T, if for any X f(x+T) = f(x) .

Constancy intervals
Intervals on which the function is either only positive or only negative are called intervals of constant sign.

f(x)>0 for x∈(x 1 , x 2)∪(x 2 , +∞), f(x)<0 при x∈(-∞,x 1)∪(x 1 , x 2)

Function continuity
The function f(x) is called continuous at the point x 0 if the limit of the function as x → x 0 is equal to the value of the function at this point, i.e. .

break points
The points at which the continuity condition is violated are called points of discontinuity of the function.

x0- breaking point.

General scheme for plotting functions

1. Find the domain of the function D(y).
2. Find the intersection points of the graph of functions with the coordinate axes.
3. Investigate the function for even or odd.
4. Investigate the function for periodicity.
5. Find intervals of monotonicity and extremum points of the function.
6. Find intervals of convexity and inflection points of the function.
7. Find the asymptotes of the function.
8. Based on the results of the study, build a graph.

Example: Explore the function and build its graph: y = x 3 - 3x
8) Based on the results of the study, we will construct a graph of the function:

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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.

Another New Year's Eve... frosty weather and snowflakes on the window pane... All this prompted me to write again about... fractals, and what Wolfram Alpha knows about it. On this occasion, there is an interesting article in which there are examples of two-dimensional fractal structures. Here we will consider more complex examples of three-dimensional fractals.

A fractal can be visually represented (described) as a geometric figure or body (meaning that both are a set, in this case, a set of points), the details of which have the same shape as the original figure itself. That is, it is a self-similar structure, considering the details of which, when magnified, we will see the same shape as without magnification. Whereas in the case of a regular geometric figure (not a fractal), when zoomed in, we will see details that have a simpler shape than the original figure itself. For example, at a sufficiently high magnification, part of an ellipse looks like a straight line segment. This does not happen with fractals: with any increase in them, we will again see the same complex shape, which with each increase will be repeated again and again.

Benoit Mandelbrot, the founder of the science of fractals, in his article Fractals and Art for Science wrote: "Fractals are geometric shapes that are as complex in their details as they are in their overall form. That is, if part of the fractal will be enlarged to the size of the whole, it will look like the whole, or exactly, or perhaps with a slight deformation.

- (Math.) A function y \u003d f (x) is called even if it does not change when the independent variable only changes sign, that is, if f (x) \u003d f (x). If f (x) = f (x), then the function f (x) is called odd. For example, y \u003d cosx, y \u003d x2 ... ...

F(x) = x is an example of an odd function. f(x) = x2 is an example of an even function. f(x) = x3 ... Wikipedia

A function that satisfies the equality f (x) = f (x). See Even and Odd Functions... Great Soviet Encyclopedia

F(x) = x is an example of an odd function. f(x) = x2 is an example of an even function. f(x) = x3 ... Wikipedia

Special functions introduced by the French mathematician E. Mathieu in 1868 when solving problems on the oscillation of an elliptical membrane. M. f. are also used in the study of the propagation of electromagnetic waves in an elliptical cylinder ... Great Soviet Encyclopedia

The "sin" request is redirected here; see also other meanings. The "sec" request is redirected here; see also other meanings. "Sine" redirects here; see also other meanings ... Wikipedia

Even and odd functions are one of its main properties, and parity 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.

Which to one degree or another were familiar to you. It was also noted there that the stock of function properties will be gradually replenished. Two new properties will be discussed in this section.

Definition 1.

The function y \u003d f (x), x є X, is called even if for any value x from the set X the equality f (-x) \u003d f (x) is true.

Definition 2.

The function y \u003d f (x), x є X, is called odd if for any value x from the set X the equality f (-x) \u003d -f (x) is true.

Prove that y = x 4 is an even function.

Solution. We have: f (x) \u003d x 4, f (-x) \u003d (-x) 4. But (-x) 4 = x 4 . Hence, for any x, the equality f (-x) = f (x), i.e. the function is even.

Similarly, it can be proved that the functions y - x 2, y \u003d x 6, y - x 8 are even.

Prove that y = x 3 is an odd function.

Solution. We have: f (x) \u003d x 3, f (-x) \u003d (-x) 3. But (-x) 3 = -x 3 . Hence, for any x, the equality f (-x) \u003d -f (x), i.e. the function is odd.

Similarly, it can be proved that the functions y \u003d x, y \u003d x 5, y \u003d x 7 are odd.

You and I have repeatedly convinced ourselves that new terms in mathematics most often have an “earthly” origin, i.e. they can be explained in some way. This is the case for both even and odd functions. See: y - x 3, y \u003d x 5, y \u003d x 7 are odd functions, while y \u003d x 2, y \u003d x 4, y \u003d x 6 are even functions. And in general, for any function of the form y \u003d x "(below we will specifically study these functions), where n is a natural number, we can conclude: if n is an odd number, then the function y \u003d x" is odd; if n is an even number, then the function y = xn is even.

There are also functions that are neither even nor odd. Such, for example, is the function y \u003d 2x + 3. Indeed, f (1) \u003d 5, and f (-1) \u003d 1. As you can see, here Hence, neither the identity f (-x) \u003d f ( x), nor the identity f(-x) = -f(x).

So, a function can be even, odd, or neither.

The study of the question of whether a given function is even or odd is usually called the study of the function for parity.

Definitions 1 and 2 deal with the values of the function at the points x and -x. This assumes that the function is defined both at the point x and at the point -x. This means that the point -x belongs to the domain of the function at the same time as the point x. If a numerical set X together with each of its elements x contains the opposite element -x, then X is called a symmetric set. Let's say (-2, 2), [-5, 5], (-oo, +oo) are symmetric sets, while )

General scheme for plotting functions

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