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

Schedule 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.

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 whether a function is even or odd

1)
; 2)
; 3)
.

Solution.

1) The function is defined when
. We'll find
.

Those.
. This means that this function is even.

2) The function is defined when

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 function of general form.

3. Study of the function for monotonicity.

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

Functions increasing (decreasing) over a certain interval are called monotonic.

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

Example 6.3. Find intervals of monotonicity of functions

1)
; 3)
.

Solution.

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

The derivative is equal to zero if
And
. The domain of definition is the number axis, divided by dots
,
at 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 increases over this interval.

2) This function is defined if
or

.

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

Thus, the domain of definition 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 over the interval
.

4. Study of the function at the extremum.

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

.

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 zero or does not exist are called critical.

5. Sufficient conditions 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 a function
equal to zero
, and the second derivative exists and is different from zero. If
, That – maximum point, if
, That – 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
– 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
– minimum points.

When passing through a point
the derivative changes sign from “+” to “–”, so
– maximum point.

,
.

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

Having solved the equation
, we'll find
And
– critical points. If the denominator
, i.e.
, then the derivative does not exist. So,
– third critical point. Let us determine the sign of the derivative in intervals.

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

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

Let's find the derivative

.

Let's find critical points:

Neighborhoods of points
do not belong to the domain of definition, therefore they are not extrema. So, let's examine the critical points
And
.

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

Let's find critical points:

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

At points
function has a minimum.

At points
the function has a maximum.

In July 2020, NASA launches an expedition to Mars. Spacecraft will deliver to Mars an electronic medium with the names of all registered expedition participants.

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Another New Year's Eve... frosty weather and snowflakes on the window glass... All this prompted me to write again about... fractals, and what Wolfram Alpha knows about it. On this occasion there is interesting article, which contains examples of two-dimensional fractal structures. Here we will look at more complex examples three-dimensional fractals.

A fractal can be visually represented (described) as a geometric figure or body (meaning that both are a set, in this case, a set of points), the details of which have the same shape as the original figure itself. That is, this is a self-similar structure, examining the details of which when magnified, we will see the same shape as without magnification. Whereas in the case of ordinary geometric figure(not a fractal), when zoomed in we will see details that have more simple form than the original figure itself. For example, at a high enough magnification, part of an ellipse looks like a straight line segment. This does not happen with fractals: with any increase in them, we will again see the same complex shape, which will be repeated again and again with each increase.

Benoit Mandelbrot, the founder of the science of fractals, wrote in his article Fractals and Art in the Name of Science: “Fractals are geometric shapes, which are equally complex in their details as in their general form. That is, if part of a fractal is enlarged to the size of the whole, it will appear as the whole, either exactly, or perhaps with a slight deformation."

. To do this, use graph paper or a graphing calculator. Select any number of numeric values ​​for the independent variable x (\displaystyle x) and plug them into the function to calculate the values ​​for the dependent variable y (\displaystyle y) . Plot the found coordinates of the points on coordinate plane, and then connect these points to graph the function.

• Substitute positive ones into the function numeric values x (\displaystyle x) and corresponding negative numeric values. For example, given a function f (x) = 2 x 2 + 1 (\displaystyle f(x)=2x^(2)+1) . Substitute the following values ​​x (\displaystyle x) into it:

Check whether the graph of the function is symmetrical about the Y axis. By symmetry we mean the mirror image of the graph about the y-axis. If the part of the graph to the right of the Y-axis (positive values ​​of the independent variable) is the same as the part of the graph to the left of the Y-axis (negative values ​​of the independent variable), the graph is symmetrical about the Y-axis. If the function is symmetrical about the y-axis, the function is even.

Check whether the graph of the function is symmetrical about the origin. The origin is the point with coordinates (0,0). Symmetry about the origin means that a positive value of y (\displaystyle y) (for a positive value of x (\displaystyle x) ) corresponds to a negative value of (\displaystyle y) (\displaystyle y) (for a negative value of x (\displaystyle x) ), and vice versa. Odd functions have symmetry about the origin.

Check if the graph of the function has any symmetry. The last type of function is a function whose graph has no symmetry, that is, there is no mirror image both relative to the ordinate axis and relative to the origin. For example, given the function .

• Substitute several positive and corresponding ones into the function negative values x (\displaystyle x) :
• According to the results obtained, there is no symmetry. The values ​​of y (\displaystyle y) for opposite values ​​of x (\displaystyle x) are not the same and are not opposite. Thus the function is neither even nor odd.
• Please note that the function f (x) = x 2 + 2 x + 1 (\displaystyle f(x)=x^(2)+2x+1) can be written as follows: f (x) = (x + 1) 2 (\displaystyle f(x)=(x+1)^(2)) . When written in this form, the function appears even because there is an even exponent. But this example proves that the type of function cannot be quickly determined if the independent variable is enclosed in parentheses. In this case, you need to open the brackets and analyze the obtained exponents.

Which were familiar to you to one degree or another. It was also noted there that the stock of function properties will be gradually replenished. Two new properties will be discussed in this section.

Definition 1.

The function y = f(x), x є X, is called even if for any value x from the set X the equality f (-x) = f (x) holds.

Definition 2.

The function y = f(x), x є X, is called odd if for any value x from the set X the equality f (-x) = -f (x) holds.

Prove that y = x 4 is an even function.

Solution. We have: f(x) = x 4, f(-x) = (-x) 4. But(-x) 4 = x 4. This means that for any x the equality f(-x) = f(x) holds, i.e. the function is even.

Similarly, it can be proven that the functions y - x 2, y = x 6, y - x 8 are even.

Prove that y = x 3 ~ an odd function.

Solution. We have: f(x) = x 3, f(-x) = (-x) 3. But (-x) 3 = -x 3. This means that for any x the equality f (-x) = -f (x) holds, i.e. the function is odd.

Similarly, it can be proven that the functions y = x, y = x 5, y = x 7 are odd.

You and I have already been convinced more than once that new terms in mathematics most often have an “earthly” origin, i.e. they can be explained somehow. This is the case with both even and odd functions. See: y - x 3, y = x 5, y = x 7 are odd functions, while y = x 2, y = x 4, y = x 6 are even functions. And in general, for any function of the form y = x" (below we will specifically study these functions), where n is a natural number, we can conclude: if n is not even number, then the function y = x" is odd; if n is an even number, then the function y = xn is even.

There are also functions that are neither even nor odd. Such, for example, is the function y = 2x + 3. Indeed, f(1) = 5, and f (-1) = 1. As you can see, here, therefore, neither the identity f(-x) = f ( x), nor the identity f(-x) = -f(x).

So, a function can be even, odd, or neither.

Studying the question of whether given function even or odd is usually called the study of a function for parity.

In definitions 1 and 2 we're talking about about the values ​​of the function at points x and -x. This assumes that the function is defined at both point x and point -x. This means that point -x belongs to the domain of definition of the function simultaneously with point x. If a numerical set X, together with each of its elements x, also contains the opposite element -x, then X is called a symmetric set. Let's say, (-2, 2), [-5, 5], (-oo, +oo) are symmetric sets, while )