Function study.

1) D(y) – Definition domain: the set of all those values ​​of the variable x. for which the algebraic expressions f(x) and g(x) make sense.

If a function is given by a formula, then the domain of definition consists of all values ​​of the independent variable for which the formula makes sense.

2) Properties of the function: even/odd, periodicity:

Odd And even functions are called whose graphs are symmetric with respect to changes in the sign of the argument.

Odd function- a function that changes its value to the opposite when the sign of the independent variable changes (symmetrical relative to the center of coordinates).

Even function- a function that does not change its value when the sign of the independent variable changes (symmetrical about the ordinate).

Neither even nor odd function (function general view) - a function that does not have symmetry. This category includes functions that do not fall under the previous 2 categories.

Functions that do not belong to any of the categories above are called neither even nor odd(or general functions).

Odd functions

Odd power where is an arbitrary integer.

Even functions

Even power where is an arbitrary integer.

Periodic function- a function that repeats its values ​​at some regular argument interval, that is, it does not change its value when adding some fixed non-zero number to the argument ( period functions) over the entire domain of definition.

3) Zeros (roots) of a function are the points where it becomes zero.

Finding the intersection point of the graph with the axis Oy. To do this you need to calculate the value f(0). Find also the points of intersection of the graph with the axis Ox, why find the roots of the equation f(x) = 0 (or make sure there are no roots).

The points at which the graph intersects the axis are called function zeros. To find the zeros of a function you need to solve the equation, that is, find those meanings of "x", at which the function becomes zero.

4) Intervals of constancy of signs, signs in them.

Intervals where the function f(x) maintains sign.

The interval of constancy of sign is the interval at every point of which the function is positive or negative.

ABOVE the x-axis.

BELOW the axle.

5) Continuity (points of discontinuity, nature of the discontinuity, asymptotes).

Continuous function- a function without “jumps”, that is, one in which small changes in the argument lead to small changes in the value of the function.

Removable Break Points

If the limit of the function exists, but the function is not defined at this point, or the limit does not coincide with the value of the function at this point:

,

then the point is called removable break point functions (in complex analysis, a removable singular point).

If we “correct” the function at the point of removable discontinuity and put , then we get a function that is continuous at a given point. Such an operation on a function is called extending the function to continuous or redefinition of the function by continuity, which justifies the name of the point as a point removable rupture.

Discontinuity points of the first and second kind

If a function has a discontinuity at a given point (that is, the limit of the function at a given point is absent or does not coincide with the value of the function at a given point), then for numerical functions there are two possible options associated with the existence of numerical functions unilateral limits:

if both one-sided limits exist and are finite, then such a point is called discontinuity point of the first kind. Removable discontinuity points are discontinuity points of the first kind;

if at least one of the one-sided limits does not exist or is not a finite value, then such a point is called point of discontinuity of the second kind.

Asymptote - straight, which has the property that the distance from a point on the curve to this direct tends to zero as the point moves away along the branch to infinity.

Vertical

Vertical asymptote - limit line .

As a rule, when determining the vertical asymptote, they look for not one limit, but two one-sided ones (left and right). This is done in order to determine how the function behaves as it approaches the vertical asymptote from different directions. For example:

Horizontal

Horizontal asymptote - straight species, subject to the existence limit

.

Inclined

Oblique asymptote - straight species, subject to the existence limits

Note: a function can have no more than two oblique (horizontal) asymptotes.

Note: if at least one of the two limits mentioned above does not exist (or is equal to ), then the oblique asymptote at (or ) does not exist.

if in item 2.), then , and the limit is found using the horizontal asymptote formula, .

6) Finding intervals of monotonicity. Find intervals of monotonicity of a function f(x)(that is, intervals of increasing and decreasing). This is done by examining the sign of the derivative f(x). To do this, find the derivative f(x) and solve the inequality f(x)0. On intervals where this inequality holds, the function f(x)increases. Where the reverse inequality holds f(x)0, function f(x) is decreasing.

Finding a local extremum. Having found the intervals of monotonicity, we can immediately determine the local extremum points where an increase is replaced by a decrease, local maxima are located, and where a decrease is replaced by an increase, local minima are located. Calculate the value of the function at these points. If a function has critical points that are not local extremum points, then it is useful to calculate the value of the function at these points as well.

Finding the largest and smallest values ​​of the function y = f(x) on a segment(continuation)

1. Find the derivative of the function: f(x). 2. Find the points at which the derivative is zero: f(x)=0x 1, x 2 ,... 3. Determine the affiliation of points X 1 ,X 2 , … segment [ a; b]: let x 1a;b, A x 2a;b .

Dependence of variable y on variable x, in which each value of x corresponds single meaning 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 build a graph 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.

Even function.

Even is a function whose sign does not change when the sign changes x.

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

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

Examples of an even function:

y=cos x

y = x 2

y = –x 2

y = x 4

y = x 6

y = x 2 + x

Explanation:
Let's take the function y = x 2 or y = –x 2 .
For any value x the function is positive. Sign x does not affect the 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 changes x.

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

The graph of an odd function is symmetrical about the origin (Fig. 2).

Examples of odd function:

y= sin x

y = x 3

y = –x 3

Explanation:

Let's take the function y = – x 3 .
All meanings at it will have a minus sign. That is a sign x influences the sign y. If the independent variable is 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 functions are even or odd. There are functions that do not obey 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 that describe these processes are called periodic functions. That is, these are functions in whose graphs there are elements that repeat at certain numerical intervals.

Converting graphs.

Verbal description of the function.

Graphic method.

The graphical method of specifying a function is the most visual and is often used in technology. IN mathematical analysis The graphical method 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 definition of this function.

A subset of the coordinate plane is a graph of a function if it has no more than one common point with any straight 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 and where it decreases. From the graph you can immediately find out some important characteristics of the function.

In general, analytical and graphical methods 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 wouldn’t even notice in the formula.

Almost any student knows the three ways to define a function that we just looked at.

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

There is such a way.

The function can be quite unambiguously specified in words.

For example, the function y=2x can be specified by the following verbal description: each real value of the argument x is associated with its double value. The rule is established, the function is specified.

Moreover, you can verbally specify a function that is extremely difficult, if not impossible, to define using 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 problematic to write this down in a formula. But it’s easy to make a sign.

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

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 - a formula, a tablet, a graph, words - does not change the essence of the matter.

Let us consider functions whose domains of definition are symmetrical with respect to the origin, i.e. for anyone X from the domain of definition 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 from its domain of definition

Example. Consider the function

It is even. Let's check it out.

For anyone X equalities are satisfied

Thus, both conditions are met, which means the function is even. Below is a graph of this function.

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

Example. Consider the function

It is odd. Let's check it out.

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

For anyone X equalities are satisfied

Thus, both conditions are met, which means the function is odd. Below is a graph of this function.

The graphs shown in the first and third figures are symmetrical about the ordinate 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?

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 Oh, 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 of the positive periods 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 .