How to prove that a function is even. The main properties of the function: even, odd, periodicity, boundedness. Basic properties of functions

Even function.

Even A function whose sign does not change when the sign is changed is called x.

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

Schedule even function symmetrical about the coordinate axis (Fig. 1).

Even function examples:

y= cos x

y = x 2

y = –x 2

y = x 4

y = x 6

y = x 2 + x

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

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

The graph of an odd function is symmetrical with respect to the origin (Fig. 2).

Examples of an odd function:

y= sin x

y = x 3

y = –x 3

Explanation:

Take the function y = - x 3 .
All values at it will have a minus sign. That is the sign x affects the sign y. If the independent variable is a 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 features are even or odd. There are functions that are not subject to 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 describing these processes are called periodic functions. That is, these are functions in whose graphs there are elements that repeat at certain numerical intervals.
















Back forward

Attention! The slide preview is for informational purposes only and may not represent the full extent of the presentation. If you are interested in this work, please download the full version.

Goals:

  • to form the concept of even and odd functions, to teach the ability to determine and use these properties when function research, plotting;
  • to develop the creative activity of students, logical thinking, the ability to compare, generalize;
  • to cultivate diligence, mathematical culture; develop communication skills .

Equipment: multimedia installation, interactive whiteboard, handouts.

Forms of work: frontal and group with elements of search and research activities.

Information sources:

 1. Algebra class 9 A.G. Mordkovich. Textbook.
2. Algebra Grade 9 A.G. Mordkovich. Task book.
3. Algebra grade 9. Tasks for learning and development of students. Belenkova E.Yu. Lebedintseva E.A.

DURING THE CLASSES

1. Organizational moment

Setting goals and objectives of the lesson.

2. Checking homework

No. 10.17 (Problem book 9th grade A.G. Mordkovich).

A) at = f(X), f(X) =

b) f (–2) = –3; f (0) = –1; f(5) = 69;

c) 1. D( f) = [– 2; + ∞)
2. E( f) = [– 3; + ∞)
3. f(X) = 0 for X ~ 0,4
4. f(X) >0 at X > 0,4 ; f(X) < 0 при – 2 < X < 0,4.
5. The function increases with X € [– 2; + ∞)
6. The function is limited from below.
7. at hire = - 3, at naib doesn't exist
8. The function is continuous.

(Did you use the feature exploration algorithm?) Slide.

2. Let's check the table that you were asked on the slide.

Fill the table

Domain

Function zeros

Constancy intervals

 Coordinates of the points of intersection of the graph with Oy

x = -5,
x = 2

х € (–5;3) U
U(2;∞)

х € (–∞;–5) U
U (–3;2)

x ∞ -5,
x ≠ 2

х € (–5;3) U
U(2;∞)

х € (–∞;–5) U
U (–3;2)

x ≠ -5,
x ≠ 2

x € (–∞; –5) U
U(2;∞)

x € (–5; 2)

3. Knowledge update

– Functions are given.
– Specify the domain of definition for each function.
– Compare the value of each function for each pair of argument values: 1 and – 1; 2 and - 2.
– For which of the given functions in the domain of definition are the equalities f(– X) = f(X), f(– X) = – f(X)? (put the data in the table) Slide

f(1) and f(– 1) f(2) and f(– 2) charts f(– X) = –f(X) f(– X) = f(X)
1. f(X) =
2. f(X) = X 3
3. f(X) = | X |
4.f(X) = 2X – 3
5. f(X) =

X ≠ 0
6. f(X)= X > –1

and not defined.

4. new material

- While doing this work, guys, we have revealed one more property of the function, unfamiliar to you, but no less important than the others - this is the evenness and oddness of the function. Write down the topic of the lesson: “Even and odd functions”, our task is to learn how to determine the even and odd functions, find out the significance of this property in the study of functions and plotting.
So, let's find the definitions in the textbook and read (p. 110) . Slide

Def. 1 Function at = f (X) defined on the set X is called even, if for any value XЄ X in progress equality f (–x) = f (x). Give examples.

Def. 2 Function y = f(x), defined on the set X is called odd, if for any value XЄ X the equality f(–х)= –f(х) is satisfied. Give examples.

Where did we meet the terms "even" and "odd"?
Which of these functions will be even, do you think? Why? Which are odd? Why?
For any function of the form at= x n, Where n is an integer, it can be argued that the function is odd for n is odd and the function is even for n- even.
– View functions at= and at = 2X– 3 is neither even nor odd, because equalities are not met f(– X) = – f(X), f(– X) = f(X)

The study of the question of whether a function is even or odd is called the study of a function for parity. Slide

Definitions 1 and 2 dealt with the values ​​of the function at x and - x, thus it is assumed that the function is also defined at the value X, and at - X.

ODA 3. If a number set together with each of its elements x contains the opposite element x, then the set X is called a symmetric set.

Examples:

(–2;2), [–5;5]; (∞;∞) are symmetric sets, and , [–5;4] are nonsymmetric.

- Do even functions have a domain of definition - a symmetric set? The odd ones?
- If D( f) is an asymmetric set, then what is the function?
– Thus, if the function at = f(X) is even or odd, then its domain of definition is D( f) is a symmetric set. But is the converse true, if the domain of a function is a symmetric set, then it is even or odd?
- So the presence of a symmetric set of the domain of definition is a necessary condition, but not a sufficient one.
– So how can we investigate the function for parity? Let's try to write an algorithm.

Slide

Algorithm for examining a function for parity

1. Determine whether the domain of the function is symmetrical. If not, then the function is neither even nor odd. If yes, then go to step 2 of the algorithm.

2. Write an expression for f(–X).

3. Compare f(–X).And f(X):

  • If f(–X).= f(X), then the function is even;
  • If f(–X).= – f(X), then the function is odd;
  • If f(–X) ≠ f(X) And f(–X) ≠ –f(X), then the function is neither even nor odd.

Examples:

Investigate the function for parity a) at= x 5 +; b) at= ; V) at= .

Solution.

a) h (x) \u003d x 5 +,

1) D(h) = (–∞; 0) U (0; +∞), symmetric set.

2) h (- x) \u003d (-x) 5 + - x5 - \u003d - (x 5 +),

3) h (- x) \u003d - h (x) \u003d\u003e function h(x)= x 5 + odd.

b) y =,

at = f(X), D(f) = (–∞; –9)? (–9; +∞), asymmetric set, so the function is neither even nor odd.

V) f(X) = , y = f(x),

1) D( f) = (–∞; 3] ≠ ; b) (∞; –2), (–4; 4]?

Option 2

1. Is the given set symmetric: a) [–2;2]; b) (∞; 0], (0; 7) ?


A); b) y \u003d x (5 - x 2). 2. Examine the function for parity:

a) y \u003d x 2 (2x - x 3), b) y \u003d

3. In fig. plotted at = f(X), for all X, satisfying the condition X? 0.
Plot the Function at = f(X), If at = f(X) is an even function.

3. In fig. plotted at = f(X), for all x satisfying x? 0.
Plot the Function at = f(X), If at = f(X) is an odd function.

Mutual check on slide.

6. Homework: №11.11, 11.21,11.22;

Proof of the geometric meaning of the parity property.

*** (Assignment of the USE option).

1. The odd function y \u003d f (x) is defined on the entire real line. For any non-negative value of the variable x, the value of this function coincides with the value of the function g( X) = X(X + 1)(X + 3)(X– 7). Find the value of the function h( X) = at X = 3.

7. Summing up

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 about 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.

In July 2020, NASA launches an expedition to Mars. spacecraft will deliver to Mars an electronic carrier with the names of all registered members of the expedition.


If this post solved your problem or you just liked it, share the link to it with your friends on social networks.

One of these code options needs to be copied and pasted into the code of your web page, preferably between the tags And or right after the tag . According to the first option, MathJax loads faster and slows down the page less. But the second option automatically tracks and loads the latest versions of MathJax. If you insert the first code, then it will need to be updated periodically. If you paste the second code, then the pages will load more slowly, but you will not need to constantly monitor MathJax updates.

The easiest way to connect MathJax is in Blogger or WordPress: in the site control panel, add a widget designed to insert third-party JavaScript code, copy the first or second version of the load code above into it, and place the widget closer to the beginning of the template (by the way, this is not necessary at all , since the MathJax script is loaded asynchronously). That's all. Now learn the MathML, LaTeX, and ASCIIMathML markup syntax and you're ready to embed math formulas into your web pages.

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 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 a 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, it is a self-similar structure, considering the details of which, when magnified, we will see the same shape as without magnification. Whereas in the case of the usual geometric figure(not a fractal), when zoomed in, we will see details that have a simpler shape than the original figure itself. For example, at a sufficiently high 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 with each increase will be repeated again and again.

Benoit Mandelbrot, the founder of the science of fractals, in his article Fractals and Art for Science wrote: "Fractals are geometric shapes that are as complex in their details as they are in their overall form. That is, if part of the fractal will be enlarged to the size of the whole, it will look like the whole, or exactly, or perhaps with a slight deformation.

Function research.

1) D(y) - Domain of definition: the set of all those values ​​of the variable x. under which the algebraic expressions f(x) and g(x) make sense.

If the 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) Function properties: even/odd, periodicity:

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

    odd function- a function that changes the value to the opposite when the sign of the independent variable changes (symmetric about the center of coordinates).

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

    Neither even nor odd function (general function) is 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 generic functions).

Odd functions

An odd power where is an arbitrary integer.

Even functions

An even power where is an arbitrary integer.

Periodic function is a function that repeats its values ​​at some regular interval of the argument, i.e., does not change its value when some fixed nonzero number is added to the argument ( period functions) over the entire domain of definition.

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

Finding the point of intersection 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 where the graph intersects the axis are called function zeros. To find the zeros of the function, you need to solve the equation, that is, find those x values, for which the function vanishes.

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

Intervals where the function f(x) retains its sign.

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

ABOVE the x-axis.

BELOW axis.

5) Continuity (points of discontinuity, character of 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 breakpoints

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

,

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

If we "correct" the function at the point of a removable discontinuity and put , then we get a function that is continuous at this point. Such an operation on a function is called extending the function to continuous or extension of the function by continuity, which justifies the name of the point, as points disposable gap.

Discontinuity points of the first and second kind

If the 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 related to the existence of numerical functions unilateral limits:

    if both one-sided limits exist and are finite, then such a point is called breaking 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 breaking point of the second kind.

Asymptote - straight, which has the property that the distance from a point of the curve to this straight tends to zero as the point moves 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

.

oblique

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 oblique asymptote at (or ) does not exist.

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

6) Finding intervals of monotonicity. Find monotonicity intervals of a function f(x) (that is, intervals of increase and decrease). 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 the intervals where this inequality is satisfied, the function f(x) increases. Where the reverse inequality holds f(x)0, function f(x) decreases.

Finding a local extremum. Having found the intervals of monotonicity, we can immediately determine the points of a local extremum where the increase is replaced by a decrease, there are local maxima, and where the decrease is replaced by an increase, local minima. 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 a function: f(x).

2. Find points where the derivative is zero: f(x)=0x 1, x 2 ,...

3. Determine the ownership of points X 1 ,X 2 , … segment [ a; b]: let x 1a;b, A x 2a;b .

4. Find function values ​​at selected points and at the ends of the segment: f(x 1), f(x 2),..., f(x a),f(x b),

5. Selection of the largest and smallest values ​​of the function from those found.

Comment. If on the segment [ a; b] there are discontinuity points, then it is necessary to calculate one-sided limits in them, and then take their values ​​into account in choosing the largest and smallest values ​​of the function.

7) Finding intervals of convexity and concavity. This is done by examining the sign of the second derivative f(x). Find the inflection points at the junctions of the convex and concavity intervals. Calculate the value of the function at the inflection points. If the function has other points of continuity (other than inflection points) at which the second derivative is equal to 0 or does not exist, then at these points it is also useful to calculate the value of the function. Finding f(x) , we solve the inequality f(x)0. On each of the solution intervals, the function will be downward convex. Solving the reverse inequality f(x)0, we find the intervals on which the function is convex upwards (that is, concave). We define inflection points as those points at which the function changes the direction of convexity (and is continuous).

Function inflection point- this is the point at which the function is continuous and when passing through which the function changes the direction of convexity.

Conditions of existence

Necessary condition for the existence of an inflection point: if the function is twice differentiable in some punctured neighborhood of the point , then either .

mob_info