What does an odd function mean. How to determine even and odd functions

Definition1. Functioning even (odd ) if together with each value of the variable
value - h.also belong
and equality is performed

Thus, the function can be even or odd only when its area of \u200b\u200bdetermining is symmetrical relative to the origin on the numeric line (the number h.and - h.at the same time belong
). For example, a function
is not even and odd, since its area of \u200b\u200bdefinition
not symmetrical about the start of coordinates.

Function
even because
symmetrical relative to the start of coordinates and.

Function
odd because
and
.

Function
is not even and odd, because though
and symmetrical on the origin of the coordinates, equality (11.1) is not performed. For example,.

An even function graph is symmetrical about the axis OUsince if the point

also belongs to schedule. Schedule of an odd function is symmetrical relative to the start of coordinates, since
belongs graphics and point
also belongs to graphics.

In proof of parity or oddness, the following statements are useful.

Theorem1. a) the sum of two even (odd) functions have an even function (odd).

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

c) The product of even and odd functions has an odd function.

d) if f.- even function on the set H., and function g. defined on a set
, then function
- even.

e) if f.- odd feature on the set H., and function g. defined on a set
and even (odd), then the function
- Even (odd).

Evidence. We prove, for example, b) and d).

b) Let
and
- even functions. Then, therefore. Similarly, the case of odd functions is considered.
and
.

d) Let f. - even function. Then.

The remaining statements of the theorem are proved similarly. Theorem is proved.

Theorem2. Any function
set H., symmetrical relative to the start of coordinates, can be represented as the sum of even and odd functions.

Evidence. Function
can be written in the form

.

Function
- even, since
, and function
- odd, because. In this way,
where
- even, and
- odd functions. Theorem is proved.

Definition2. Function
called periodic if there is a number
, such that in any
numbers
and
also belong areas of definition
and equality are performed

Such a number T.called period functions
.

From Definition 1 it follows that if T.- Period of function
, then the number - T.also is a function period
(since when replacing T.on the - T.equality is preserved). With the help of the method of mathematical induction, you can show that if T.- Period of function f., that I.
is also a period. It follows that if the function has a period, then it has infinitely many periods.

Definition3. The smallest of the positive periods of the function is called it basic period.

Theorem3. If T.- the main period of the function f., the remaining periods are painted to him.

Evidence. Suppose nasty, that is, there is a period functions f. (\u003e 0), not multiple T.. Then, share on the T.with the remnant, we get
where
. therefore

i.e - Period of function f., and
, and this contradicts what T.- the main period of the function f.. The assertion of the theorem follows from the resulting contradiction. Theorem is proved.

It is well known that trigonometric functions are periodic. The main period
and
raven
,
and
. Find a function of the function
. Let be
- The period of this function. Then

(as
.

elijah
.

Value T.defined from the first equality can not be a period because it depends on h.. is a function OT. h., not a constant number. The period is determined from the second equality:
. Periods are infinitely a lot, when
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.- rational number, then
and
are rational numbers with rational h.and irrational with irrational h.. therefore

with any rational number T.. Therefore, any rational number T.is a period of Dirichlet function. It is clear that there is no main period in this function, since there are positive rational numbers, how much are close to zero (for example, a rational settlement to choose n.how much is close to zero).

Theorem4. If the function f. set on the set H.and has a period T., and function g. set on the set
, then a complex function
also has a period T..

Evidence. We have, therefore

that is, the statement of the theorem is proved.

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

Definition4. Functions that are not periodic are called non-periodic .

The dependence of the variable Y from the variable X, at which each value x corresponds to the only value of y is called the function. For the designation use the entry y \u003d f (x). Each function has a number of basic properties, such as monotony, parity, frequency, and others.

Consider a readiness of parity.

The function y \u003d f (x) is called even if it satisfies the following two conditions:

2. The value of the function at the point x, which belongs to the function definition area should be equal to the function value at the point. That is, for any point x, the following equality F (X) \u003d F (-X) should be performed from the function of determining the function.

Schedule even function

If you construct an even function graph, it will be symmetrical with respect to the OU axis.

For example, the function y \u003d x ^ 2 is even. Check it. The definition area of \u200b\u200bthe entire number axis, which means it is symmetric about the point O.

Take an arbitrary x \u003d 3. f (x) \u003d 3 ^ 2 \u003d 9.

f (-X) \u003d (- 3) ^ 2 \u003d 9. Consequently, f (x) \u003d f (-x). Thus, we have both conditions, it means that the function is even. Below is a graph of the function y \u003d x ^ 2.

The figure shows that the schedule is symmetrical with respect to the OU axis.

Chart of odd function

The function y \u003d f (x) is called odd if it satisfies the following two conditions:

1. The field of definition of this function must be symmetrical with respect to the point of O., if some point A belongs to the function of determining the function, then the corresponding point -a must also belong to the definition area of \u200b\u200bthe specified function.

2. For any point x, the following equality f (x) \u003d -f (x) should be performed from the function definition area.

The schedule of an odd function is symmetrical about the point of the beginning of the coordinates. For example, the function y \u003d x ^ 3 is odd. Check it. The definition area of \u200b\u200bthe entire number axis, which means it is symmetric about the point O.

Take arbitrary x \u003d 2. f (x) \u003d 2 ^ 3 \u003d 8.

f (-X) \u003d (- 2) ^ 3 \u003d -8. Consequently, f (x) \u003d -f (x). Thus, we have both conditions, it means the functions are odd. Below is a graph of the function y \u003d x ^ 3.

The figure clearly shows that the odd function y \u003d x ^ 3 is symmetrical relative to the start of coordinates.

The function is called even (odd) if equality is performed for anyone

.

An even function graph is symmetrical about the axis
.

The schedule of an odd function is symmetrical on the start of the coordinates.

Example 6.2. Explore the parity or oddness of the function

1)
; 2)
; 3)
.

Decision.

1) the function is determined when
. Find
.

Those.
. So this function is even.

2) the function is determined when

Those.
. Thus, this function is odd.

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

,
. Therefore, the function is neither even or odd. We call it a common type function.

3. Investigation of the function on monotony.

Function
it is called increasing (decreasing) at some interval, if in this interval each greater value of the argument corresponds to a greater (smaller) value of the function.

The functions of increasing (decreasing) are called monotonous at some interval.

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

Example 6.3.. Find intervals of functions monotony

1)
; 3)
.

Decision.

1) This function is determined on the entire numeric axis. Find a derivative.

The derivative is zero if
and
. The definition area is the numerical axis, divided by points
,
at intervals. Determine the sign of the derivative in each interval.

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

In the interval
the derivative is positive, therefore, the function at this interval increases.

2) This function is defined if
or

.

Determine the sign of square three declections in each interval.

Thus, the field definition area

Find a derivative
,
, if a
.
, but
. 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. Study of the function to extremum.

Point
called a maximum point (minimum) functions
if there is such a neighborhood point that for all
inequality is performed from this neighborhood

.

Maximum points and a minimum of functions are called extremum points.

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

The points in which the derivative is equal to zero or is not called critical.

5. Sufficient conditions for the existence of extremum.

Rule 1.. If during the transition (from left to right) through a critical point derivative
changes the sign from "+" to "-", then at the point function
has a maximum; If with "-" on "+", then a minimum; if a
does not change the sign, then the extremum is not.

Rule 2.. Let in the point
first derivative function
equal to zero.
, and the second derivative exists and is different from zero. If a
T. - Maximum point if
T. - point minimum function.

Example 6.4 . Explore the maximum and minimum function:

1)
; 2)
; 3)
;

4)
.

Decision.

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

Find a derivative
and solve equation
.
.OTSyud
- Critical points.

Determine the sign of the derivative in the intervals,
.

When moving through points
and
the derivative changes the sign from "-" to "+", so according to rule 1
- Minimum points.

When switching through the point
the derivative changes the sign from "+" to "-", therefore
- Maximum point.

,
.

2) The function is defined and continuous in the interval
. Find a derivative
.

Deciding equation
We find
and
- Critical points. If the denominator
.
, the derivative does not exist. So,
- Third critical point. Determine the sign of the derivative in the intervals.

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

3) the function is defined and continuous if
. for
.

Find a derivative

.

We will find critical points:

Neighborhoods
do not belong to the definition area, so they are not t. Extremum. So, we investigate critical points
and
.

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

We will find critical points:

We find the second derivative
and define her sign at points

At points
the function has a minimum.

At points
the function has a maximum.

Which in one degree or another were familiar to you. There was also noticed that the stock of the properties of functions would be gradually replenished. About two new properties and will be discussed in this paragraph.

Definition 1.

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

Definition 2.

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

Prove that y \u003d x 4 is an even function.

Decision. We have: f (x) \u003d x 4, f (s) \u003d (s) 4. But (s) 4 \u003d x 4. So, for any x, the equality f (s) \u003d f (x) is performed, i.e. The function is even.

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

Prove that y \u003d x 3 ~ an odd feature.

Decision. We have: F (x) \u003d x 3, f (s) \u003d (s) 3. But (s) 3 \u003d -kh 3. So, for any x, the equality F (s) \u003d -f (x) is performed, 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.

We have already been convinced of the fact that new terms in mathematics most often have "earthly" origin, i.e. They can somehow explain them. This is the case with even, and with odd functions. See: y - x 3, y \u003d x 5, y \u003d x 7 - odd functions, while y \u003d x 2, y \u003d x 4, y \u003d x 6 - even functions. In general, for any functions of the type y \u003d x "(below we will specifically, we will 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 an odd; If N is an even number, then the function y \u003d xn is even.

There are also functions that are neither even or odd. Such is, for example, the function y \u003d 2x + 3. In fact, f (1) \u003d 5, and F (-1) \u003d 1. As you can see, it means that no identity f (-x) \u003d f ( x), nor the identity F (s) \u003d -f (x).

So, the function can be even, odd, as well as so neither the other.

Studying the question of whether a given function is even or odd, usually referred to the study of functions for parity.

In definitions 1 and 2, we are talking about the values \u200b\u200bof the function at points x and -x. Thus, it is assumed that the function is also defined at the point x, and at the point. This means that point -H belongs the field of determining the function simultaneously with the point x. If the numeric set x together with each element x contains the opposite element, then x is called a symmetric set. Let's say (-2, 2), [-5, 5], (-oo, + oo) - symmetric sets, while)