How to understand the evenness and oddness of a function. Graph of even and odd functions

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 shown below graphs of functions?

Advantage graphic task is its visibility. You can immediately see how the function behaves, where it increases and where it decreases. From the graph you can immediately recognize some important characteristics functions.

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 define 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 the sign is easy to make.

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?

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 values ​​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. This 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 straight 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)

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.

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Methods for specifying a function

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

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

Using this table, you can see that for the argument value −1 the function value −3 will correspond; and the value x=2 will correspond to y=0, etc. It is also important to know that each argument value in the table corresponds to only one function value.

More functions can be specified using graphs. Using a graph, it is established which value of the function correlates with a certain value 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 of definition. Such a function will be symmetrical about the Oy axis.

The function is odd function, when f(-x)=-f(x) for any x from the domain of definition. Such a function will be symmetric about the origin O (0;0) .

The function is not even, neither odd and is called general function, when it does not have symmetry about the axis or origin.

Let us examine the following function for parity:

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

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

This means that the function f(x)=3x^(3)-7x^(7) is odd.

Periodic function

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

Repeating the graph of a function on any segment of the x-axis that has length T.

The intervals where the function is positive, that is, f(x) > 0, are segments of the abscissa axis that correspond to the points of the function graph lying above the abscissa axis.

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

Intervals where the function is negative, that is, 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))

Limited function

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

An example of a function bounded from 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 when there is 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 is 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 entire number axis, since \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 then, when a larger value of x corresponds to a larger value of the function y=f(x) . It follows that taking two arbitrary values ​​of the argument x_(1) and x_(2) from the interval under consideration, with x_(1) > x_(2) , the result 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 corresponds to a smaller value of the function y(x) . It follows that, taking from the interval under consideration two arbitrary values ​​of the argument x_(1) and x_(2) , and x_(1) > x_(2) , the result will be y(x_(1))< y(x_{2}) .

Function Roots It is customary to call the points at which the function F=y(x) intersects the abscissa axis (they are obtained by solving the equation y(x)=0).

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

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

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

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

Extrema of the function

Minimum point of the function y=f(x) is usually called a point x=x_(0) whose neighborhood will have other points (except for the point x=x_(0)), and for them the inequality f(x) > f will then be satisfied (x_(0)) . y_(min) - designation of the function at the min point.

The maximum point of the function y=f(x) is usually called a point x=x_(0) whose neighborhood will have other points (except for the point x=x_(0)), and for them the inequality f(x) will then be satisfied< f(x^{0}) . y_{max} - обозначение функции в точке max.

Prerequisite

According to Fermat’s theorem: f"(x)=0 when the function f(x) that is differentiable at the point x_(0) will have an extremum at this point.

Sufficient condition

1. When the derivative changes sign 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 a function on an interval

Calculation steps:

1. The derivative f"(x) is sought;
2. Stationary and critical points of the function are found and those belonging to the segment are selected;
3. The values ​​of the function f(x) are found at stationary and critical points and ends of the segment. The smaller of the results obtained will be lowest value functions, and more - the largest.

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 not built even function squared, we get even.

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.