1 x even or odd. Even and odd functions. Periodic functions

Graphs of even and not even function have the following features:

If a function is even, then its graph is symmetrical about the ordinate. If a function is odd, then its graph is symmetrical about the origin.

Solution. Consider the function: \(f\left(x \right)=\left|x \right|\) and substitute the opposite \(-x \) instead of \(x \). As a result of simple transformations we get: $$f\left(-x \right)=\left|-x \right|=\left|x \right|=f\left(x \right)$$ In other words, if replace the argument with the opposite sign, the function will not change.

This means that this function is even, and its graph will be symmetrical with respect to the ordinate axis ( vertical axis). The graph of this function is shown in the figure on the left. This means that when constructing a graph, you can only draw half, and the second part (to the left of the vertical axis, draw symmetrically to the right part). By determining the symmetry of a function before starting to plot its graph, you can greatly simplify the process of constructing or studying the function. If it is difficult to perform a general check, you can do it simpler: substitute the same values ​​of different signs into the equation. For example -5 and 5. If the function values ​​turn out to be the same, then we can hope that the function will be even. From a mathematical point of view, this approach is not entirely correct, but from a practical point of view it is convenient. To increase the reliability of the result, you can substitute several pairs of such opposite values.

Solution. Let's check the same as in the previous example: $$f\left(-x \right)=x\left|-x \right|=-x\left|x \right|=-f\left(x \right) $$ This means that the original function is odd (the sign of the function has changed to the opposite).

Conclusion: the function is symmetrical about the origin. You can build only one half, and draw the second symmetrically. This kind of symmetry is more difficult to draw. This means that you are looking at the chart from the other side of the sheet, and even upside down. Or you can do this: take the drawn part and rotate it around the origin 180 degrees counterclockwise.

Solution. Let's perform the same check for sign change as in the previous two examples. $$f\left(-x \right)=\left(-x \right)^3+\left(-x \right)^2=-x^2+x^2$$ As a result, we get that: $$f\left(-x \right)\not=f\left(x \right),f\left(-x \right)\not=-f\left(x \right)$$ And this means, that the function is neither even nor odd.

Conclusion: the function is not symmetrical either with respect to the origin or the center of the coordinate system. This happened because it is the sum of two functions: even and odd. The same situation will happen if you subtract two different functions. But multiplication or division will lead to a different result. For example, the product of an even and an odd function produces an odd function. Or the quotient of two odd numbers leads to an even function.

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

Let's determine 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 of definition, the corresponding values ​​of y (function) turn out to be equal.

Let's give a more strict 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 point) also lies in this scope,

• f(-x) = f(x).

From the above definition follows the condition necessary for the domain of definition of such a function, 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 above, therefore, the conclusion follows: the even function has a form symmetrical with respect to the ordinate axis (Oy).

How to determine the parity of a function in practice?

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

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

Let's check the parity of the function h(x)=11^x-11^(-x). Following the same algorithm, we get that h(-x) = 11^(-x) -11^x. Taking out the minus, in the end we have
h(-x)=-(11^x-11^(-x))=- h(x). Therefore, 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 adding similar functions, they get an even one;
• 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 multiplying odd and even functions, an odd one is obtained;
• as a result of dividing odd and even functions, an odd one is obtained;
• the derivative of such a function is odd;
• If you square an odd function, you get an even one.

The parity of a function can be used to solve equations.

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

This is also successfully used to solve non-standard tasks with parameter.

For example, is there any value of the parameter a for which the equation 2x^6-x^4-ax^2=1 will 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 given equation won't change. It follows that if a certain number is its root, then the opposite number is also the root. The conclusion is obvious: the roots of an equation that are different from zero are included in the set of its solutions in “pairs”.

It is clear that the number itself is not 0, 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 this equation contains solutions “in pairs”. Let's check if 0 is a root. When we substitute it into the equation, we get 2=2. Thus, in addition to “paired” ones, 0 is also a root, which proves their odd number.

Definition 1. The function is called even (odd ), if together with each variable value
meaning - X also belongs
and the equality holds

Thus, a function can be even or odd only if its domain of definition is symmetrical about the origin of coordinates on the number line (number X And - X belong at the same time
). For example, the function
is neither even nor odd, since its domain of definition
not symmetrical about the origin.

Function
even, because
symmetrical about the origin and.

Function
odd, because
And
.

Function
is not even and odd, since although
and is symmetrical with respect to the origin, equalities (11.1) are not satisfied. For example,.

The graph of an even function is symmetrical about the axis OU, because if the point

also belongs to the schedule. The graph of an odd function is symmetrical about the origin, since if
belongs to the graph, then the point
also belongs to the schedule.

When proving whether a function is even or odd, the following statements are useful.

Theorem 1. a) The sum of two even (odd) functions is an even (odd) function.

b) The product of two even (odd) functions is an even function.

c) The product of an even and odd function is an odd function.

d) If f– even function on the set X, and the function g defined on the set
, then the function
– even.

d) If f– odd function on the set X, and the function g defined on the set
and even (odd), then the function
– even (odd).

Proof. Let us prove, for example, b) and d).

b) Let
And
– even functions. Then, therefore. The case of odd functions is treated similarly
And
.

d) Let f is an even function. Then.

The remaining statements of the theorem can be proved in a similar way. The theorem has been proven.

Theorem 2. Any function
, defined on the set X, symmetrical about the origin, can be represented as a sum of even and odd functions.

Proof. Function
can be written in the form

.

Function
– even, because
, and the function
– odd, because. Thus,
, Where
– even, and
– odd functions. The theorem has been proven.

Definition 2. Function
called periodic , if there is a number
, such that for any
numbers
And
also belong to the domain of definition
and the equalities are satisfied

Such a number T called period functions
.

From Definition 1 it follows that if T– period of the function
, then the number – T Same is the period of the function
(since when replacing T on - T equality is maintained). Using the method of mathematical induction it can be shown that if T– period of the function f, then
, is also a period. It follows that if a function has a period, then it has infinitely many periods.

Definition 3. The smallest positive period of a function is called its main period.

Theorem 3. If T– main period of the function f, then the remaining periods are multiples of it.

Proof. Let us assume the opposite, that is, that there is a period functions f (>0), not multiple T. Then, dividing on T with the remainder, we get
, Where
. That's why

that is – period of the function f, and
, and this contradicts the fact that T– main period of the function f. The statement of the theorem follows from the resulting contradiction. The theorem has been proven.

It is well known that trigonometric functions are periodic. Main period
And
equals
,
And
. Let's find the period of the function
. Let
- the period of this function. Then

(because
.

oror
.

Meaning T, determined from the first equality, cannot be a period, since it depends on X, i.e. is a function of X, and not a constant number. The period is determined from the second equality:
. There are infinitely many periods, with
the smallest positive period is obtained at
:
. This is the main period of the function
.

An example of a more complex periodic function is the Dirichlet function

Note that if T is a rational number, then
And
are rational numbers for rational X and irrational when irrational X. That's why

for any rational number T. Therefore, any rational number T is the period of the Dirichlet function. It is clear that this function does not have a main period, since there are positive rational numbers, arbitrarily close to zero (for example, a rational number can be made a choice n arbitrarily close to zero).

Theorem 4. If the function f defined on the set X and has a period T, and the function g defined on the set
, then a complex function
also has a period T.

Proof. We have, therefore

that is, the statement of the theorem is proven.

For example, since cos x has a period
, then the functions
have a period
.

Definition 4. Functions that are not periodic are called non-periodic .

Function is one of the most important mathematical concepts. Function - variable dependency at from variable x, if each value X matches a single value at. Variable X called the independent variable or argument. Variable at called the dependent variable. All values ​​of the independent variable (variable x) form the domain of definition of the function. All values ​​that the dependent variable takes (variable y), form the range of values ​​of the function.

Function graph call the set of all points of the coordinate plane, the abscissas of which are equal to the values ​​of the argument, and the ordinates are equal to the corresponding values ​​of the function, that is, the values ​​of the variable are plotted along the abscissa axis x, and the values ​​of the variable are plotted along the ordinate axis y. To graph a function, you need to know the properties of the function. The main properties of the function will be discussed below!

To build a graph of a function, we recommend using our program - Graphing functions online. If you have any questions while studying the material on this page, you can always ask them on our forum. Also on the forum they will help you solve problems in mathematics, chemistry, geometry, probability theory and many other subjects!

Basic properties of functions.

1) Function domain and function range.

The domain of a function is the set of all valid valid argument values x(variable x), for which the function y = f(x) determined.
The range of a function is the set of all real values y, which the function accepts.

IN elementary mathematics functions are studied only on the set of real numbers.

2) Function zeros.

Values X, at which y=0, called function zeros. These are the abscissas of the points of intersection of the function graph with the Ox axis.

3) Intervals of constant sign of a function.

Intervals of constant sign of a function are such intervals of values x, on which the function values y either only positive or only negative are called intervals of constant sign of the function.

4) Monotonicity of the function.

An increasing function (in a certain interval) is a function for which higher value the argument from this interval corresponds to a larger value of the function.

A decreasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a smaller value of the function.

5) Even (odd) function.

An even function is a function whose domain of definition is symmetrical with respect to the origin and for any X f(-x) = f(x). The graph of an even function is symmetrical about the ordinate.

An odd function is a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality is true f(-x) = - f(x). The graph of an odd function is symmetrical about the origin.

Even function
1) The domain of definition is symmetrical with respect to the point (0; 0), that is, if the point a belongs to the domain of definition, then the point -a also belongs to the domain of definition.
2) For any value x f(-x)=f(x)
3) The graph of an even function is symmetrical about the Oy axis.

Odd function has the following properties:
1) The domain of definition is symmetrical about the point (0; 0).
2) for any value x, belonging to the domain of definition, the equality f(-x)=-f(x)
3) The graph of an odd function is symmetrical with respect to the origin (0; 0).

Not every function is even or odd. Functions general view are neither even nor odd.

6) Limited and unlimited functions.

A function is called bounded if there is a positive number M such that |f(x)| ≤ M for all values ​​of x. If such a number does not exist, then the function is unlimited.

7) Periodicity of the function.

A function f(x) is periodic if there is a non-zero number T such that for any x from the domain of definition of the function the following holds: f(x+T) = f(x). This smallest number is called the period of the function. All trigonometric functions are periodic. (Trigonometric formulas).

Function f is called periodic if there is a number such that for any x from the domain of definition the equality f(x)=f(x-T)=f(x+T). T is the period of the function.

Every periodic function has infinite set periods. In practice, the smallest positive period is usually considered.

The values ​​of a periodic function are repeated after an interval equal to the period. This is used when constructing graphs.

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

Take a closer look at the parity property.

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

2. The value of the function at point x, belonging to the domain of definition of the function, must be equal to the value of the function at point -x. That is, for any point x, from the domain of definition of the function the following equality must be satisfied: f(x) = f(-x).

Graph of an even function

If you plot a graph of an even function, it will be symmetrical about the Oy 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 it is symmetrical about point O.

Let's take an arbitrary x=3. f(x)=3^2=9.

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

The figure shows that the graph is symmetrical about the Oy 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 definition of a given function must be symmetrical with respect to point O. That is, if some point a belongs to the domain of definition of the function, then the corresponding point -a must also belong to the domain of definition of the given function.

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

The graph of an odd function is symmetrical with respect to point O - the origin of coordinates. 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 it is symmetrical about point O.

Let's take an arbitrary x=2. f(x)=2^3=8.

f(-x)=(-2)^3=-8. Therefore f(x) = -f(x). Thus, both conditions are met, which means 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 about the origin.