Short. The purpose of ND filters is to reduce the amount of light entering the...
Chart conversion.
Verbal description of the function.
Graphic way.
The graphical way of specifying a function is the most illustrative and is often used in engineering. In mathematical analysis, the graphical way 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 the given function.
A subset of the coordinate plane is a graph of some function if it has at most one common point with any 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, where it decreases. From the graph, you can immediately find out some important characteristics of the function.
In general, analytical and graphical ways 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 won’t notice in the formula.
Almost any student knows the three ways to define a function that we have just covered.
Let's try to answer the question: "Are there other ways to define a function?"
There is such a way.
A function can be quite unambiguously defined in words.
For example, the function y=2x can be defined by the following verbal description: each real value of the argument x is assigned its doubled value. The rule is set, the function is set.
Moreover, it is possible to specify a function verbally, which is extremely difficult, if not impossible, to specify by 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. Etc. It is difficult to write this down in a formula. But the table is easy to make.
The method of verbal description is a rather rarely used method. But sometimes it happens.
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 - by a formula, tablet, graph, words - does not change the essence of the matter.
Consider functions whose domains of definition are symmetrical with respect to the origin of coordinates, i.e. for anyone X out of scope 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 out of its domain
Example. Consider the function 
She is even. Let's check it out.
For anyone X the equalities
Thus, both conditions are satisfied for us, which means that the function is even. Below is a graph of this function.
Definition. The function f is called odd, if for any X out of its domain 
Example. Consider the function 
She 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 (0; 0).
For anyone X the equalities
Thus, both conditions are satisfied for us, which means that the function is odd. Below is a graph of this function.
The graphs shown in the first and third figures are symmetrical about the y-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?
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 in the study of functions, plotting graphs;
- 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, |
х € (–5;3) U |
х € (–∞;–5) U |
||
 |
x ∞ -5, |
х € (–5;3) U |
х € (–∞;–5) U |
||
x ≠ -5, |
x € (–∞; –5) U |
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. one 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 fulfilled. 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= ; in) at= .
Decision.
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.
in) 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).
a) y \u003d x 2 (2x - x 3), b) y \u003d
Plot the Function at = f(X), if at = f(X) is an even function.
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
Graphs of even and odd functions have the following features:
If a function is even, then its graph is symmetrical about the y-axis. If a function is odd, then its graph is symmetrical about the origin.
Example. Plot the function \(y=\left|x \right|\).Decision. Consider the function: \(f\left(x \right)=\left|x \right|\) and substitute \(x \) for the opposite \(-x \). As a result of simple transformations, we get: $$f\left(-x \right)=\left|-x \right|=\left|x \right|=f\left(x \right)$$ In other words, if replace the argument with the opposite sign, the function will not change.
This means that this function is even, and its graph will be symmetrical about the y-axis (vertical axis). The graph of this function is shown in the figure on the left. This means that when plotting a graph, you can only draw half, and the second part (to the left of the vertical axis, draw already symmetrically to the right side). By determining the symmetry of a function before starting to plot its graph, you can greatly simplify the process of constructing or studying a function. If it is difficult to perform a check in a general form, you can do it easier: substitute the same values \u200b\u200bof different signs into the equation. For example -5 and 5. If the values of the function are the same, then we can hope that the function will be even. From a mathematical point of view, this approach is not entirely correct, but from a practical point of view, it is convenient. To increase the reliability of the result, you can substitute several pairs of such opposite values.
Example. Plot the function \(y=x\left|x \right|\).
Decision. Let's check the same as in the previous example: $$f\left(-x \right)=x\left|-x \right|=-x\left|x \right|=-f\left(x \right) $$ This means that the original function is odd (the sign of the function is reversed).
Conclusion: the function is symmetrical with respect to the origin. You can build only one half, and draw the other half symmetrically. This symmetry is more difficult to draw. This means that you are looking at the chart from the other side of the sheet, and even turned upside down. And you can also do this: take the drawn part and rotate it around the origin by 180 degrees counterclockwise.
Example. Plot the function \(y=x^3+x^2\).
Decision. Let's perform the same sign change check as in the previous two examples. $$f\left(-x \right)=\left(-x \right)^3+\left(-x \right)^2=-x^2+x^2$$ $$f\left(-x \right)\not=f\left(x \right),f\left(-x \right)\not=-f\left(x \right)$$ Which means that the function is neither even nor odd.
Conclusion: the function is not symmetrical either about the origin or about the center of the coordinate system. This happened because it is the sum of two functions: even and odd. The same situation will be if you subtract two different functions. But multiplication or division will lead to a different result. For example, the product of an even and an odd function gives an odd one. Or the quotient of two odd leads to an even function.
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 exist, 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 the 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 be x 1a;b, a x 2a;b . |
