Real processes occurring in nature can be described using functions. So, we can distinguish two main types of the flow of processes that are opposite to each other - these are gradual or continuous and spasmodic(an example would be a ball falling and rebounding). But if there are discontinuous processes, then there are special means for their description. For this purpose, functions that have discontinuities, jumps are put into circulation, that is, in different parts of the numerical line, the function behaves according to different laws and, accordingly, is given by different formulas. The concepts of discontinuity points and removable discontinuity are introduced.

Surely you have already seen functions defined by several formulas, depending on the values ​​of the argument, for example:

y \u003d (x - 3, with x\u003e -3;
(-(x - 3), for x< -3.

Such functions are called piecewise or piecewise. Sections of the number line with different job formulas, let's call constituents domain. The union of all components is the domain of the piecewise function. Those points that divide the domain of a function into components are called boundary points. Formulas that define a piecewise function on each constituent domain of definition are called incoming functions. Graphs of piecewise given functions are obtained as a result of combining parts of graphs built on each of the partition intervals.

Exercises.

Construct graphs of piecewise functions:

1) (-3, with -4 ≤ x< 0,
f(x) = (0, for x = 0,
(1, at 0< x ≤ 5.

The graph of the first function is a straight line passing through the point y = -3. It originates at the point with coordinates (-4; -3), goes parallel to the abscissa axis to the point with coordinates (0; -3). The graph of the second function is a point with coordinates (0; 0). The third graph is similar to the first - it is a straight line passing through the point y \u003d 1, but already in the area from 0 to 5 along the Ox axis.

Answer: figure 1.

2) (3 if x ≤ -4,
f(x) = (|x 2 - 4|x| + 3| if -4< x ≤ 4,
(3 - (x - 4) 2 if x > 4.

Consider each function separately and plot its graph.

So, f(x) = 3 is a straight line parallel to the Ox axis, but it needs to be drawn only in the area where x ≤ -4.

Graph of the function f(x) = |x 2 – 4|x| + 3| can be obtained from the parabola y \u003d x 2 - 4x + 3. Having built its graph, the part of the figure that lies above the Ox axis must be left unchanged, and the part that lies under the abscissa axis must be displayed symmetrically relative to the Ox axis. Then symmetrically display the part of the graph where
x ≥ 0 about the Oy axis for negative x. The graph obtained as a result of all transformations is left only in the area from -4 to 4 along the abscissa.

The graph of the third function is a parabola, the branches of which are directed downwards, and the vertex is at the point with coordinates (4; 3). The drawing is depicted only in the area where x > 4.

Answer: figure 2.

3) (8 - (x + 6) 2 if x ≤ -6,
f(x) = (|x 2 – 6|x| + 8| if -6 ≤ x< 5,
(3 if x ≥ 5.

The construction of the proposed piecewise given function is similar to the previous paragraph. Here, the graphs of the first two functions are obtained from parabola transformations, and the graph of the third is a straight line parallel to Ox.

Answer: figure 3.

4) Plot the function y = x – |x| + (x – 1 – |x|/x) 2 .

Solution. The domain of this function is all real numbers except zero. Let's open the module. To do this, consider two cases:

1) For x > 0, we get y = x - x + (x - 1 - 1) 2 = (x - 2) 2 .

2) For x< 0 получим y = x + x + (x – 1 + 1) 2 = 2x + x 2 .

Thus, we have a piecewise given function:

y = ((x - 2) 2 , for x > 0;
( x 2 + 2x, for x< 0.

The graphs of both functions are parabolas, the branches of which are directed upwards.

Answer: figure 4.

5) Plot the function y = (x + |x|/x – 1) 2 .

Solution.

It is easy to see that the domain of the function is all real numbers except zero. After expanding the module, we get a piecewise given function:

1) For x > 0, we get y = (x + 1 - 1) 2 = x 2 .

2) For x< 0 получим y = (x – 1 – 1) 2 = (x – 2) 2 .

Let's rewrite.

y \u003d (x 2, for x\u003e 0;
((x – 2) 2 , for x< 0.

The graphs of these functions are parabolas.

Answer: figure 5.

6) Is there a function whose graph on the coordinate plane has a common point with any line?

Solution.

Yes, there is.

An example would be the function f(x) = x 3 . Indeed, the graph of the cubic parabola intersects with the vertical line x = a at the point (a; a 3). Now let the straight line be given by the equation y = kx + b. Then the equation
x 3 - kx - b \u003d 0 has a real root x 0 (since a polynomial of odd degree always has at least one real root). Therefore, the graph of the function intersects with the straight line y \u003d kx + b, for example, at the point (x 0; x 0 3).

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Textbook: Algebra grade 8, edited by A. G. Mordkovich.

Lesson type: Discovery of new knowledge.

Goals:

for the teacher goals are fixed in each stage of the lesson;

for student:

Personal Goals:

• Learn to clearly, accurately, competently express your thoughts in oral and written speech, to understand the meaning of the task;
• Learn to apply acquired knowledge and skills to solving new problems;
• Learn to control the process and the result of their activities;

Meta-objective goals:

In cognitive activity:

• The development of logical thinking and speech, the ability to logically substantiate one's judgments, to carry out simple systematizations;
• Learn to put forward hypotheses when solving problems, understand the need to test them;
• Apply knowledge in a standard situation, learn how to independently perform tasks;
• To carry out the transfer of knowledge to a changed situation, to see the task in the context of a problematic situation;

In information and communication activities:

• Learn to conduct a dialogue, recognize the right to a different opinion;

In reflective activity:

• Learn to anticipate the possible consequences of your actions;
• Learn to eliminate the causes of difficulties.

Subject goals:

• Learn what is a piecewise given function;
• Learn to set a piecewise given function analytically according to its graph;

During the classes

1. Self-determination to learning activities

Purpose of the stage:

• include students in learning activities;
• determine the content of the lesson: we continue to repeat the topic of numerical functions.

Organization of the educational process at stage 1:

T: What did we do in the previous lessons?

D: We repeated the topic of numerical functions.

T: Today we will continue to repeat the topic of the previous lessons, and also today we should find out what new things we can learn about this topic.

2. Updating knowledge and fixing difficulties in activities

Purpose of the stage:

• update the educational content necessary and sufficient for the perception of new material: recall the formulas of numerical functions, their properties and methods of construction;
• to update the mental operations necessary and sufficient for the perception of new material: comparison, analysis, generalization;
• to fix an individual difficulty in activity, demonstrating the insufficiency of existing knowledge at a personally significant level: setting a piecewise given function analytically, as well as building its graph.

Organization of the educational process at stage 2:

T: There are five numerical functions on the slide. Determine their type.

1) fractional-rational;

2) quadratic;

3) irrational;

4) function with module;

5) power.

T: Name the formulas corresponding to them.

3) ;

4) ;

T: Let's discuss what role each coefficient plays in these formulas?

D: The variables “l” and “m” are responsible for shifting the graphs of these functions to the left - right and up - down, respectively, the coefficient “k” in the first function determines the position of the hyperbola branches: k>0 - the branches are in the I and III quarters, k< 0 - во II и IV четвертях, а коэффициент “а” определяет направление ветвей параболы: а>0 - branches are directed upwards, and< 0 - вниз).

2. slide 2

U: Set analytically the functions whose graphs are shown in the figures. (considering that they are moving y=x 2). The teacher writes the answers on the board.

D: 1) );

2);

3. slide 3

U: Set analytically the functions whose graphs are shown in the figures. (considering that they are moving). The teacher writes the answers on the board.

4. slide 4

U: Using the previous results, set analytically the functions whose graphs are shown in the figures.

3. Identification of the causes of difficulties and setting the goal of the activity

Purpose of the stage:

• organize communicative interaction, during which the distinctive property of the task that caused difficulty in educational activities is revealed and fixed;
• agree on the purpose and topic of the lesson.

Organization of the educational process at stage 3:

Q: What is causing you trouble?

D: Pieces of graphs are provided on the screen.

T: What is the purpose of our lesson?

D: To learn how to analytically define pieces of functions.

T: State the topic of the lesson. (Children try to formulate the topic on their own. The teacher clarifies it. Topic: Piecewise given function.)

4. Building a project for getting out of a difficulty

Purpose of the stage:

• organize communicative interaction to build a new mode of action that eliminates the cause of the identified difficulty;
• establish a new way of doing things.

Organization of the educational process at stage 4:

T: Let's read the assignment carefully again. What results are asked to be used as an aid?

D: Previous, i.e. the ones written on the board.

T: Maybe these formulas are already the answer to this task?

D: No, because. these formulas define quadratic and rational functions, and their pieces are shown on the slide.

T: Let's discuss what intervals of the x-axis correspond to the pieces of the first function?

U: Then the analytical way of specifying the first function looks like: if

Q: What needs to be done to complete a similar task?

D: Write down the formula and determine what intervals of the x-axis correspond to the pieces of this function.

5. Primary consolidation in external speech

Purpose of the stage:

• fix the studied educational content in external speech.

Organization of the educational process at stage 5:

7. Inclusion in the knowledge system and repetition

Purpose of the stage:

• practice the skills of using new content in conjunction with previously learned.

Organization of the educational process at stage 7:

Y: Set analytically the function, the graph of which is shown in the figure.

8. Reflection of activities in the lesson

Purpose of the stage:

• to fix the new content learned in the lesson;
• evaluate their own activities in the classroom;
• thank classmates who helped to get the result of the lesson;
• fix unresolved difficulties as directions for future learning activities;
• discuss and write down homework.

Organization of the educational process at stage 8:

T: What did we learn in class today?

D: With a piecewise given function.

T: What work did we learn to do today?

D: Set this type of function analytically.

T: Raise your hand, who understood the topic of today's lesson? (Discuss the problems with the rest of the children).

Homework

• No. 21.12(a, c);
• No. 21.13(a, c);
• №22.41;
• №22.44.

Continuity and plotting piecewise functions is a complex topic. It is better to learn how to build graphs directly in a practical lesson. Here, the study on continuity is mainly shown.

It is known that elementary function(see p. 16) is continuous at all points where it is defined. Therefore, discontinuity in elementary functions is possible only at points of two types:

a) at points where the function is "overridden";

b) at points where the function does not exist.

Accordingly, only such points are checked for continuity during the study, as shown in the examples.

For non-elementary functions, the study is more difficult. For example, a function (the integer part of a number) is defined on the entire number line, but suffers a break at each integer x. Questions like this are outside the scope of this guide.

Before studying the material, you should repeat from a lecture or textbook what (what kind) break points are.

Investigation of piecewise given functions for continuity

Function set piecewise, if it is given by different formulas in different parts of the domain of definition.

The main idea in the study of such functions is to find out if the function is defined at the points where it is redefined, and how. Then it is checked whether the values ​​of the function to the left and to the right of such points are the same.

Example 1 Let us show that the function
continuous.

Function
is elementary and therefore continuous at the points at which it is defined. But, obviously, it is defined at all points. Therefore, it is continuous at all points, including at
, as required by the condition.

The same is true for the function
, and at
it is continuous.

In such cases, continuity can only be broken where the function is redefined. In our example, this is the point
. Let's check it, for which we find the limits on the left and right:

The limits on the left and right are the same. It remains to be seen:

a) whether the function is defined at the point itself
;

b) if so, does it match?
with limit values ​​on the left and right.

By condition, if
, then
. That's why
.

We see that (all are equal to the number 2). This means that at the point
the function is continuous. So the function is continuous on the entire axis, including the point
.

Solution Notes

a) It did not play a role in the calculations, substitute we are in a specific number formula
or
. This is usually important when dividing by an infinitesimal value is obtained, as it affects the sign of infinity. Here
and
responsible only for function selection;

b) as a rule, designations
and
are equal, the same applies to the designations
and
(and is true for any point, not just for
). In what follows, for brevity, we use notations of the form
;

c) when the limits on the left and on the right are equal, to test for continuity, in fact, it remains to see whether one of the inequalities lax. In the example, this turned out to be the 2nd inequality.

Example 2 We investigate the continuity of the function
.

For the same reasons as in Example 1, continuity can only be broken at the point
. Let's check:

The limits on the left and right are equal, but at the point itself
the function is not defined (inequalities are strict). It means that
- dot repairable gap.

"Removable discontinuity" means that it is enough either to make any of the inequalities non-strict, or to invent for a separate point
function, the value of which at
is -5, or simply indicate that
so that the whole function
became continuous.

Answer: dot
– break point.

Remark 1. In the literature, a removable gap is usually considered a special case of a gap of the 1st kind, however, students are more often understood as a separate type of gap. In order to avoid discrepancies, we will adhere to the 1st point of view, and we will specifically stipulate the “irremovable” gap of the 1st kind.

Example 3 Check if the function is continuous

At the point

The limits on the left and right are different:
. Whether or not the function is defined
(yes) and if so, what is equal to (is equal to 2), point
–point of irremovable discontinuity of the 1st kind.

At the point
going on final jump(from 1 to 2).

Answer: dot

Remark 2. Instead of
and
usually write
and
respectively.

Available question: how are the functions different

and
,

and also their charts? Right answer:

a) 2nd function is not defined at point
;

b) on the graph of the 1st function, the point
"painted over", on the graph 2 - no ("punctured point").

Dot
where the graph ends
, is not shaded in both graphs.

It is more difficult to study functions that are defined differently on three plots.

Example 4 Is the function continuous?
?

Just like in examples 1 - 3, each of the functions
,
and is continuous on the entire numerical axis, including the section on which it is given. The gap is only possible at the point
or (and) at the point
where the function is overridden.

The task is divided into 2 subtasks: to investigate the continuity of the function

and
,

moreover, the point
not of interest to the function
, and the point
- for the function
.

1st step. Checking the point
and function
(we do not write the index):

The limits match. By condition,
(if the limits on the left and on the right are equal, then the function is actually continuous when one of the inequalities is not strict). So at the point
the function is continuous.

2nd step. Checking the point
and function
:

Because the
, dot
is a discontinuity point of the 1st kind, and the value
(and whether it exists at all) no longer matters.

Answer: the function is continuous at all points except the point
, where there is an unrecoverable discontinuity of the 1st kind - a jump from 6 to 4.

Example 5 Find function break points
.

We act in the same way as in example 4.

1st step. Checking the point
:

a)
, because to the left of
the function is constant and equal to 0;

b) (
is an even function).

The limits are the same, but
the function is not defined by the condition, and it turns out that
– break point.

2nd step. Checking the point
:

a)
;

b)
- the value of the function does not depend on the variable.

The limits are different: , dot
is the point of irremovable discontinuity of the 1st kind.

Answer:
– break point,
is a point of irremovable discontinuity of the 1st kind, at other points the function is continuous.

Example 6 Is the function continuous?
?

Function
determined at
, so the condition
becomes a condition
.

On the other hand, the function
determined at
, i.e. at
. So the condition
becomes a condition
.

It turns out that the condition must be satisfied
, and the domain of definition of the entire function is the segment
.

The functions themselves
and
are elementary and therefore continuous at all points at which they are defined—in particular, and for
.

It remains to check what happens at the point
:

a)
;

Because the
, see if the function is defined at the point
. Yes, the 1st inequality is not strict with respect to
, and that's enough.

Answer: the function is defined on the interval
and continuous on it.

More complex cases, when one of the constituent functions is non-elementary or not defined at any point in its segment, are beyond the scope of the tutorial.

NF1. Plot function graphs. Pay attention to whether the function is defined at the point at which it is redefined, and if so, what is the value of the function (the word " if» is omitted in the function definition for brevity):

1) a)
b)
in)
G)

2) a)
b)
in)
G)

3) a)
b)
in)
G)

4) a)
b)
in)
G)

Example 7 Let
. Then on the site
build a horizontal line
, and on the site
build a horizontal line
. In this case, the point with coordinates
"gouged out" and the dot
"painted over". At the point
a discontinuity of the 1st kind (“jump”) is obtained, and
.

NF2. Investigate for continuity the functions defined differently on 3 intervals. Plot the graphs:

1) a)
b)
in)

G)
e)
e)

2) a)
b)
in)

G)
e)
e)

3) a)
b)
in)

G)
e)
e)

Example 8 Let
. Location on
build a straight line
, for which we find
and
. Connecting the dots
and
segment. We do not include the points themselves, since for
and
the function is not defined by the condition.

Location on
and
circle the OX axis (on it
), but the points
and
"knocked out". At the point
we obtain a removable discontinuity, and at the point
– discontinuity of the 1st kind (“jump”).

NF3. Plot the function graphs and make sure they are continuous:

1) a)
b)
in)

G)
e)
e)

2) a)
b)
in)

G)
e)
e)

NF4. Make sure the functions are continuous and build their graphs:

1) a)
b)
in)

2 a)
b)
in)

3) a)
b)
in)

NF5. Plot function graphs. Pay attention to continuity:

1) a)
b)
in)

G)
e)
e)

2) a)
b)
in)

G)
e)
e)

3) a)
b)
in)

G)
e)
e)

4) a)
b)
in)

G)
e)
e)

5) a)
b)
in)

G)
e)
e)

NF6. Plot graphs of discontinuous functions. Note the value of the function at the point where the function is redefined (and whether it exists):

1) a)
b)
in)

G)
e)
e)

2) a)
b)
in)

G)
e)
e)

3) a)
b)
in)

G)
e)
e)

4) a)
b)
in)

G)
e)
e)

5) a)
b)
in)

G)
e)
e)

NF7. Same task as in NF6:

1) a)
b)
in)

G)
e)
e)

2) a)
b)
in)

G)
e)
e)

3) a)
b)
in)

G)
e)
e)

4) a)
b)
in)

G)
e)
e)

Analytical definition of a function

Function %%y = f(x), x \in X%% given in an explicit analytical way, if a formula is given that indicates the sequence of mathematical operations that must be performed with the argument %%x%% to get the value %%f(x)%% of this function.

Example

• %% y = 2 x^2 + 3x + 5, x \in \mathbb(R)%%;
• %% y = \frac(1)(x - 5), x \neq 5%%;
• %% y = \sqrt(x), x \geq 0%%.

So, for example, in physics, with uniformly accelerated rectilinear motion, the speed of a body is determined by the formula t%% is written as: %% s = s_0 + v_0 t + \frac(a t^2)(2) %%.

Piecewise Defined Functions

Sometimes the function under consideration can be defined by several formulas that operate in different parts of the domain of its definition, in which the function argument changes. For example: $$ y = \begin(cases) x ^ 2,~ if~x< 0, \\ \sqrt{x},~ если~x \geq 0. \end{cases} $$

Functions of this kind are sometimes called constituent or piecewise. An example of such a function is %%y = |x|%%

Function scope

If a function is specified in an explicit analytical way using a formula, but the scope of the function in the form of a set %%D%% is not specified, then by %%D%% we will always mean the set of values ​​of the argument %%x%% for which this formula makes sense . So for the function %%y = x^2%%, the domain of definition is the set %%D = \mathbb(R) = (-\infty, +\infty)%%, since the argument %%x%% can take any values ​​on number line. And for the function %%y = \frac(1)(\sqrt(1 - x^2))%%, the domain of definition will be the set of values ​​%%x%% satisfying the inequality %%1 - x^2 > 0%%, m .e. %%D = (-1, 1)%%.

Benefits of Explicit Analytic Function Definition

Note that the explicit analytical way of specifying a function is quite compact (the formula, as a rule, takes up little space), easily reproduced (the formula is easy to write down), and is most adapted to performing mathematical operations and transformations on functions.

Some of these operations - algebraic (addition, multiplication, etc.) - are well known from the school mathematics course, others (differentiation, integration) will be studied in the future. However, this method is not always clear, since the nature of the dependence of the function on the argument is not always clear, and sometimes cumbersome calculations are required to find the values ​​of the function (if they are necessary).

Implicit function specification

The function %%y = f(x)%% is defined in an implicit analytical way, if the relation $$F(x,y) = 0 is given, ~~~~~~~~~~(1)$$ relating the values ​​of the function %%y%% and the argument %%x%%. If given argument values, then to find the value of %%y%% corresponding to a particular value of %%x%%, it is necessary to solve the equation %%(1)%% with respect to %%y%% at that particular value of %%x%%.

Given a value of %%x%%, the equation %%(1)%% may have no solution or more than one solution. In the first case, the specified value %%x%% is not in the scope of the implicit function, and in the second case it specifies multivalued function, which has more than one value for a given argument value.

Note that if the equation %%(1)%% can be explicitly solved with respect to %%y = f(x)%%, then we obtain the same function, but already defined in an explicit analytical way. So, the equation %%x + y^5 - 1 = 0%%

and the equality %%y = \sqrt(1 - x)%% define the same function.

Parametric function definition

When the dependence of %%y%% on %%x%% is not given directly, but instead the dependences of both variables %%x%% and %%y%% on some third auxiliary variable %%t%% are given in the form

$$ \begin(cases) x = \varphi(t),\\ y = \psi(t), \end(cases) ~~~t \in T \subseteq \mathbb(R), ~~~~~ ~~~~~(2) $$they talk about parametric the method of setting the function;

then the auxiliary variable %%t%% is called a parameter.

If it is possible to exclude the parameter %%t%% from the equations %%(2)%%, then they come to a function given by an explicit or implicit analytical dependence of %%y%% on %%x%%. For example, from the relations $$ \begin(cases) x = 2 t + 5, \\ y = 4 t + 12, \end(cases), ~~~t \in \mathbb(R), $$ except for the parameter % %t%% we get the dependence %%y = 2 x + 2%%, which sets a straight line in the %%xOy%% plane.

Graphical way

An example of a graphical definition of a function

The above examples show that the analytical way of defining a function corresponds to its graphic image, which can be considered as a convenient and visual form of describing a function. Sometimes used graphic way defining a function when the dependence of %%y%% on %%x%% is given by a line on the %%xOy%% plane. However, for all its clarity, it loses in accuracy, since the values ​​of the argument and the corresponding values ​​of the function can be obtained from the graph only approximately. The resulting error depends on the scale and accuracy of measuring the abscissa and ordinate of the individual points of the graph. In the future, we will assign the role of the graph of the function only to illustrate the behavior of the function, and therefore we will restrict ourselves to the construction of "sketches" of graphs that reflect the main features of the functions.

Tabular way

Note tabular way function assignments, when some argument values ​​and their corresponding function values ​​are placed in a table in a certain order. This is how the well-known tables of trigonometric functions, tables of logarithms, etc. are constructed. In the form of a table, the relationship between the quantities measured in experimental studies, observations, and tests is usually presented.

The disadvantage of this method is the impossibility of directly determining the values ​​of the function for the values ​​of the argument that are not included in the table. If there is confidence that the argument values ​​not presented in the table belong to the domain of the considered function, then the corresponding function values ​​can be calculated approximately using interpolation and extrapolation.

Example

x	3	5.1	10	12.5
y	9	23	80	110

Algorithmic and verbal ways of specifying functions

The function can be set algorithmic(or programmatic) in a way that is widely used in computer calculations.

Finally, it may be noted descriptive(or verbal) a way of specifying a function, when the rule for matching the values ​​of the function to the values ​​of the argument is expressed in words.

For example, the function %%[x] = m~\forall (x \in )