- (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

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

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

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 distribution 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

A function is called even (odd) if for any and the equality

.

The graph of an even function is symmetrical about the axis
.

The graph of an odd function is symmetrical about the origin.

Example 6.2. Examine for even or odd functions

1)
; 2)
; 3)
.

Solution.

1) The function is defined with
. Let's find
.

Those.
. Means, given function is even.

2) The function is defined for

Those.
. Thus, this function is odd.

3) the function is defined for , i.e. for

,
. Therefore, the function is neither even nor odd. Let's call it a general function.

3. Investigation of a function for monotonicity.

Function
is called increasing (decreasing) on ​​some interval if in this interval each larger value of the argument corresponds to a larger (smaller) value of the function.

Functions increasing (decreasing) on ​​some interval are called monotonic.

If the function
differentiable on the interval
and has a positive (negative) derivative
, then the function
increases (decreases) in this interval.

Example 6.3. Find intervals of monotonicity of functions

1)
; 3)
.

Solution.

1) This function is defined on the entire number axis. Let's find the derivative.

The derivative is zero if
and
. Domain of definition - numerical axis, divided by points
,
for intervals. Let us determine the sign of the derivative in each interval.

In the interval
the derivative is negative, the function decreases on this interval.

In the interval
the derivative is positive, therefore, the function is increasing on this interval.

2) This function is defined if
or

.

We determine the sign of the square trinomial in each interval.

Thus, the scope of the function

Let's find the derivative
,
, if
, i.e.
, but
. Let us determine the sign of the derivative in the intervals
.

In the interval
the derivative is negative, therefore, the function decreases on the interval
. In the interval
the derivative is positive, the function increases on the interval
.

4. Investigation of a function for an extremum.

Dot
is called the maximum (minimum) point of the function
, if there is such a neighborhood of the point that for everyone
this neighborhood satisfies the inequality

.

The maximum and minimum points of a function are called extremum points.

If the function
at the point has an extremum, then the derivative of the function at this point is equal to zero or does not exist (a necessary condition for the existence of an extremum).

The points at which the derivative is equal to zero or does not exist are called critical.

5. Sufficient conditions for the existence of an extremum.

Rule 1. If during the transition (from left to right) through the critical point derivative
changes sign from "+" to "-", then at the point function
has a maximum; if from "-" to "+", then the minimum; if
does not change sign, then there is no extremum.

Rule 2. Let at the point
first derivative of the function
zero
, and the second derivative exists and is nonzero. If a
, then is the maximum point, if
, then is the minimum point of the function.

Example 6.4 . Explore the maximum and minimum functions:

1)
; 2)
; 3)
;

4)
.

Solution.

1) The function is defined and continuous on the interval
.

Let's find the derivative
and solve the equation
, i.e.
.from here
are critical points.

Let us determine the sign of the derivative in the intervals ,
.

When passing through points
and
the derivative changes sign from “–” to “+”, therefore, according to rule 1
are the minimum points.

When passing through a point
derivative changes sign from "+" to "-", so
is the maximum point.

,
.

2) The function is defined and continuous in the interval
. Let's find the derivative
.

By solving the equation
, find
and
are critical points. If the denominator
, i.e.
, then the derivative does not exist. So,
is the third critical point. Let us determine the sign of the derivative in intervals.

Therefore, the function has a minimum at the point
, maximum at points
and
.

3) A function is defined and continuous if
, i.e. at
.

Let's find the derivative

.

Let's find the critical points:

Neighborhoods of points
do not belong to the domain of definition, so they are not extremum t. So let's explore the critical points
and
.

4) The function is defined and continuous on the interval
. We use rule 2. Find the derivative
.

Let's find the critical points:

Let's find the second derivative
and determine its sign at the points

At points
function has a minimum.

At points
function has a maximum.

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.

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

Thus, a function can be even or odd only when its domain of definition is symmetrical with respect to the origin of coordinates on the real line (numbers X and - X simultaneously belong
). For example, the function
is neither even nor odd, since its domain of definition
not symmetrical about the origin.

Function
even, because
symmetrical with respect to the origin of coordinates and.

Function
odd because
and
.

Function
is neither even nor odd, since although
and is symmetric 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, since if the point

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

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 an odd function is an odd function.

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

e) If f is an 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
are even functions. Then, therefore. The case of odd functions is considered similarly
and
.

d) Let f is an even function. Then.

The other assertions of the theorem are proved similarly. The theorem has been proven.

Theorem 2. Any function
, defined on the set X, which is symmetric with respect to the origin, can be represented as the sum of an even and an odd function.

Proof. Function
can be written in the form

.

Function
is even, since
, and the function
is odd because. In this way,
, where
- even, and
is an odd function. The theorem has been proven.

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

Such a number T called period functions
.

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

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

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

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

that is – function period f, and
, which contradicts the fact that T is the main period of the function f. The assertion of the theorem follows from the obtained contradiction. The theorem has been proven.

It is well known that trigonometric functions are periodic. Main period
and
equals
,
and
. Find the period of the function
. Let
is the period of this function. Then

(because
.

ororor
.

Meaning T, determined from the first equality, cannot be a period, since it depends on X, i.e. is a function of X, not a constant number. The period is determined from the second equality:
. There are infinitely many periods
the smallest positive period is obtained when
:
. 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 under 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 has no main period, since there are positive rational numbers arbitrarily close to zero (for example, a rational number can be made by choosing n arbitrarily close to zero).

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

Proof. We have therefore

that is, the assertion of the theorem is proved.

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 .

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Ways to set a function

Let the function be given by the formula: y=2x^(2)-3 . By assigning any value to the independent variable x , you can use this formula to calculate the corresponding values ​​of the dependent variable y . For example, if x=-0.5 , then using the formula, we get that the corresponding value of y is y=2 \cdot (-0.5)^(2)-3=-2.5 .

Given any value taken by the x argument in the formula y=2x^(2)-3 , only one function value can be calculated that corresponds to it. The function can be represented as a table:

x	−2	−1	0	1	2	3
y	−4	−3	−2	−1	0	1

Using this table, you can figure out that for the value of the argument -1, the value of the function -3 will correspond; and the value x=2 will correspond to y=0, and so on. It is also important to know that each argument value in the table corresponds to only one function value.

More functions can be set using graphs. With the help of the graph, it is established which value of the function correlates with a certain value of x. Most often, this will be an approximate value of the function.

Even and odd function

The function is even function, when f(-x)=f(x) for any x from the domain. Such a function will be symmetrical about the Oy axis.

The function is odd function when f(-x)=-f(x) for any x in the domain. Such a function will be symmetrical about the origin O (0;0) .

The function is not even, nor odd and called function general view when it does not have symmetry about the axis or origin.

We examine the following function for parity:

f(x)=3x^(3)-7x^(7)

D(f)=(-\infty ; +\infty) with a symmetrical domain of definition about the origin. f(-x)= 3 \cdot (-x)^(3)-7 \cdot (-x)^(7)= -3x^(3)+7x^(7)= -(3x^(3)-7x^(7))= -f(x).

Hence, the function f(x)=3x^(3)-7x^(7) is odd.

Periodic function

The function y=f(x) , in the domain of which f(x+T)=f(x-T)=f(x) is true for any x, is called periodic function with period T \neq 0 .

Repetition of the graph of the function on any segment of the abscissa axis, which has length T .

Intervals where the function is positive, that is, f (x) > 0 - segments of the abscissa axis, which correspond to the points of the graph of the function that lie above the abscissa axis.

f(x) > 0 on (x_(1); x_(2)) \cup (x_(3); +\infty)

Gaps where the function is negative, i.e. f(x)< 0 - отрезки оси абсцисс, которые отвечают точкам графика функции, лежащих ниже оси абсцисс.

f(x)< 0 на (-\infty; x_(1)) \cup (x_(2); x_(3))

Function limitation

bounded from below it is customary to call a function y=f(x), x \in X when there exists a number A for which the inequality f(x) \geq A holds for any x \in X .

An example of a function bounded below: y=\sqrt(1+x^(2)) since y=\sqrt(1+x^(2)) \geq 1 for any x .

bounded from above a function y=f(x), x \in X is called if there exists a number B for which the inequality f(x) \neq B holds for any x \in X .

An example of a function bounded below: y=\sqrt(1-x^(2)), x \in [-1;1] since y=\sqrt(1+x^(2)) \neq 1 for any x \in [-1;1] .

Limited it is customary to call a function y=f(x), x \in X when there exists a number K > 0 for which the inequality \left | f(x) \right | \neq K for any x \in X .

Example limited function: y=\sin x is limited on the whole number line, because \left | \sin x \right | \neq 1.

Increasing and decreasing function

It is customary to speak of a function that increases on the interval under consideration as increasing function when a larger value of x will correspond to a larger value of the function y=f(x) . From here it turns out that taking from the considered interval two arbitrary values ​​of the argument x_(1) and x_(2) , and x_(1) > x_(2) , it will be y(x_(1)) > y(x_(2)) .

A function that decreases on the interval under consideration is called decreasing function when a larger value of x will correspond to a smaller value of the function y(x) . From here it turns out that taking from the considered interval two arbitrary values ​​of the argument x_(1) and x_(2) , and x_(1) > x_(2) , it will be y(x_(1))< y(x_{2}) .

Function roots it is customary to name the points at which the function F=y(x) intersects the abscissa axis (they are obtained as a result of solving the equation y(x)=0 ).

a) If an even function increases for x > 0, then it decreases for x< 0

b) When an even function decreases for x > 0, then it increases for x< 0

c) When an odd function increases for x > 0, then it also increases for x< 0

d) When an odd function decreases for x > 0, then it will also decrease for x< 0

Function extremes

Function minimum point y=f(x) it is customary to call such a point x=x_(0) , in which its neighborhood will have other points (except for the point x=x_(0) ), and for them then the inequality f(x) > f (x_(0)) . y_(min) - designation of the function at the point min.

Function maximum point y=f(x) it is customary to call such a point x=x_(0) , in which its neighborhood will have other points (except for the point x=x_(0) ), and then the inequality f(x) will be satisfied for them< f(x^{0}) . y_{max} - обозначение функции в точке max.

Necessary condition

According to Fermat's theorem: f"(x)=0, then when the function f(x) , which is differentiable at the point x_(0) , an extremum will appear at this point.

Sufficient condition

1. When the sign of the derivative changes from plus to minus, then x_(0) will be the minimum point;
2. x_(0) - will be a maximum point only when the derivative changes sign from minus to plus when passing through the stationary point x_(0) .

The largest and smallest value of the function on the interval

Calculation steps:

1. Looking for derivative f"(x) ;
2. Stationary and critical points of the function are found and those belonging to the interval are chosen;
3. The values ​​of the function f(x) are found at the stationary and critical points and ends of the segment. The smallest of the results will be the smallest value of the function, and more - greatest.