Y cube root of x graph. Function y = third root of x, its properties and graph

The basic properties of the power function are given, including formulas and properties of the roots. Derivative, integral, expansion in power series and representation through complex numbers of a power function.

Definition

Definition
Power function with exponent p is the function f (x) = x p, the value of which at point x is equal to the value of the exponential function with base x at point p.
In addition, f (0) = 0 p = 0 for p > 0 .

For natural values of the exponent, the power function is the product of n numbers equal to x:
.
It is defined for all valid .

For positive rational values of the exponent, the power function is the product of n roots of degree m of the number x:
.
For odd m, it is defined for all real x. For even m, the power function is defined for non-negative ones.

For negative , the power function is determined by the formula:
.
Therefore, it is not defined at the point.

For irrational values of the exponent p, the power function is determined by the formula:
,
where a is an arbitrary positive number not equal to one: .
When , it is defined for .
When , the power function is defined for .

Continuity. A power function is continuous in its domain of definition.

Properties and formulas of power functions for x ≥ 0

Here we will consider the properties of the power function for not negative values argument x. As stated above, for certain values of the exponent p, the power function is also defined for negative values of x. In this case, its properties can be obtained from the properties of , using even or odd. These cases are discussed and illustrated in detail on the page "".

A power function, y = x p, with exponent p has the following properties:
(1.1) defined and continuous on the set
at ,
at ;
(1.2) has many meanings
at ,
at ;
(1.3) strictly increases with ,
strictly decreases as ;
(1.4) at ;
at ;
(1.5) ;
(1.5*) ;
(1.6) ;
(1.7) ;
(1.7*) ;
(1.8) ;
(1.9) .

Proof of properties is given on the page “Power function (proof of continuity and properties)”

Roots - definition, formulas, properties

Definition
Root of a number x of degree n is the number that when raised to the power n gives x:
.
Here n = 2, 3, 4, ... - natural number, greater than one.

You can also say that the root of a number x of degree n is the root (i.e. solution) of the equation
.
Note that the function is the inverse of the function.

Square root of x is a root of degree 2: .

Cube root of x is a root of degree 3: .

Even degree

For even powers n = 2 m, the root is defined for x ≥ 0 . A formula that is often used is valid for both positive and negative x:
.
For square root:
.

The order in which the operations are performed is important here - that is, first squaring is performed, resulting in a non-negative number, and then the root is extracted from it (you can extract from a non-negative number Square root). If we changed the order: , then for negative x the root would be undefined, and with it the entire expression would be undefined.

Odd degree

For odd powers, the root is defined for all x:
;
.

Properties and formulas of roots

The root of x is a power function:
.
When x ≥ 0 the following formulas apply:
;
;
, ;
.

These formulas can also be applied for negative values of variables. You just need to make sure that the radical expression of even powers is not negative.

Private values

The root of 0 is 0: .
Root 1 is equal to 1: .
The square root of 0 is 0: .
The square root of 1 is 1: .

Example. Root of roots

Let's look at an example of a square root of roots:
.
Let's transform the inner square root using the formulas above:
.
Now let's transform the original root:
.
So,
.

y = x p for different values of the exponent p.

Here are graphs of the function for non-negative values of the argument x. Graphs of a power function defined for negative values of x are given on the page “Power function, its properties and graphs"

Inverse function

The inverse of a power function with exponent p is a power function with exponent 1/p.

If, then.

Derivative of a power function

Derivative of nth order:
;

Deriving formulas > > >

Integral of a power function

P ≠ - 1 ;
.

Power series expansion

At - 1 < x < 1 the following decomposition takes place:

Expressions using complex numbers

Consider the function of the complex variable z:
f (z) = z t.
Let us express the complex variable z in terms of the modulus r and the argument φ (r = |z|):
z = r e i φ .
We represent the complex number t in the form of real and imaginary parts:
t = p + i q .
We have:

Next, we take into account that the argument φ is not uniquely defined:
,

Let's consider the case when q = 0 , that is, the exponent is a real number, t = p. Then
.

If p is an integer, then kp is an integer. Then, due to the periodicity of trigonometric functions:
.
That is, the exponential function with an integer exponent, for a given z, has only one value and is therefore unambiguous.

If p is irrational, then the products kp for any k do not produce an integer. Since k runs through an infinite series of values k = 0, 1, 2, 3, ..., then the function z p has infinitely many values. Whenever the argument z is incremented 2π(one turn), we move to a new branch of the function.

If p is rational, then it can be represented as:
, Where m, n- integers that do not contain common divisors. Then
.
First n values, with k = k 0 = 0, 1, 2, ... n-1, give n different meanings kp:
.
However, subsequent values give values that differ from the previous ones by an integer. For example, when k = k 0+n we have:
.
Trigonometric functions, whose arguments differ by values that are multiples of 2π, have equal values. Therefore, with a further increase in k, we obtain the same values of z p as for k = k 0 = 0, 1, 2, ... n-1.

Thus, an exponential function with a rational exponent is multivalued and has n values (branches). Whenever the argument z is incremented 2π(one turn), we move to a new branch of the function. After n such revolutions we return to the first branch from which the countdown began.

In particular, a root of degree n has n values. As an example, consider the nth root of the real positive number z = x. In this case φ 0 = 0 , z = r = |z| = x, .
.
So, for a square root, n = 2 ,
.
For even k, (- 1 ) k = 1. For odd k, (- 1 ) k = - 1.
That is, the square root has two meanings: + and -.

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.

Guys, we continue to study power functions. The topic of today's lesson will be the function - the cubic root of x. What is a cube root? The number y is called a cube root of x (root of the third degree) if the equality is satisfied. Designated by:, where x is the radical number, 3 is the exponent.

As we can see, the cube root can also be extracted from negative numbers. It turns out that our root exists for all numbers. The third root of a negative number is negative number. When raised to an odd power, the sign is preserved; the third power is odd. Let's check the equality: Let. Let us raise both expressions to the third power. Then or In the notation of roots we obtain the desired identity.

Guys, let's now build a graph of our function. 1) Domain set real numbers. 2) The function is odd, since Next we will consider our function at x 0, then we will display the graph relative to the origin. 3) The function increases as x 0. For our function, a larger value of the argument corresponds to a larger value of the function, which means increase. 4) The function is not limited from above. In fact, from any large number we can calculate the third root, and we can go up to infinity, finding everything large values argument. 5) When x 0 the smallest value is 0. This property is obvious.

Let's construct our graph of the function over the entire domain of definition. Remember that our function is odd. Properties of the function: 1) D(y)=(-;+) 2) Odd function. 3) Increases by (-;+) 4) Unlimited. 5) There is no minimum or maximum value. 6) The function is continuous on the entire number line. 7) E(y)= (-;+). 8) Convex downward by (-;0), convex upward by (0;+).

Example. Draw a graph of the function and read it. Solution. Let's construct two graphs of functions on the same coordinate plane, taking into account our conditions. For x-1 we build a graph of the cube root, for x-1 we build a graph linear function. 1) D(y)=(-;+) 2) The function is neither even nor odd. 3) Decreases by (-;-1), increases by (-1;+) 4) Unlimited from above, limited from below. 5) Greatest value No. Lowest value equals minus one. 6) The function is continuous on the entire number line. 7) E(y)= (-1;+)

Lesson and presentation on the topic: "Power functions. Cubic root. Properties of the cubic root"

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Definition of a power function - cube root

Guys, we continue to study power functions. Today we will talk about the "Cubic root of x" function.
What is a cube root?
The number y is called a cube root of x (root of the third degree) if the equality $y^3=x$ holds.
Denoted as $\sqrt(x)$, where x is a radical number, 3 is an exponent.
$\sqrt(27)=3$; $3^3=$27.
$\sqrt((-8))=-2$; $(-2)^3=-8$.
As we can see, the cube root can also be extracted from negative numbers. It turns out that our root exists for all numbers.
The third root of a negative number is equal to a negative number. When raised to an odd power, the sign is preserved; the third power is odd.

Let's check the equality: $\sqrt((-x))$=-$\sqrt(x)$.
Let $\sqrt((-x))=a$ and $\sqrt(x)=b$. Let's raise both expressions to the third power. $–x=a^3$ and $x=b^3$. Then $a^3=-b^3$ or $a=-b$. Using the notation for roots we obtain the desired identity.

Properties of cubic roots

a) $\sqrt(a*b)=\sqrt(a)*\sqrt(6)$.
b) $\sqrt(\frac(a)(b))=\frac(\sqrt(a))(\sqrt(b))$.

Let's prove the second property. $(\sqrt(\frac(a)(b)))^3=\frac(\sqrt(a)^3)(\sqrt(b)^3)=\frac(a)(b)$.
We found that the number $\sqrt(\frac(a)(b))$ cubed is equal to $\frac(a)(b)$ and then equals $\sqrt(\frac(a)(b))$, which and needed to be proven.

Guys, let's build a graph of our function.
1) Domain of definition is the set of real numbers.
2) The function is odd, since $\sqrt((-x))$=-$\sqrt(x)$. Next, consider our function for $x≥0$, then display the graph relative to the origin.
3) The function increases when $x≥0$. For our function, a larger value of the argument corresponds to a larger value of the function, which means increase.
4) The function is not limited from above. In fact, from an arbitrarily large number we can calculate the third root, and we can move upward indefinitely, finding ever larger values of the argument.
5) For $x≥0$ the smallest value is 0. This property is obvious.
Let's build a graph of the function by points at x≥0.

Let's construct our graph of the function over the entire domain of definition. Remember that our function is odd.

Function properties:
1) D(y)=(-∞;+∞).
2) Odd function.
3) Increases by (-∞;+∞).
4) Unlimited.
5) There is no minimum or maximum value.

7) E(y)= (-∞;+∞).
8) Convex downward by (-∞;0), convex upward by (0;+∞).

Examples of solving power functions

Examples
1. Solve the equation $\sqrt(x)=x$.
Solution. Let's construct two graphs on the same coordinate plane $y=\sqrt(x)$ and $y=x$.

As you can see, our graphs intersect at three points.
Answer: (-1;-1), (0;0), (1;1).

2. Construct a graph of the function. $y=\sqrt((x-2))-3$.
Solution. Our graph is obtained from the graph of the function $y=\sqrt(x)$, parallel transfer two units to the right and three units down.

3. Graph the function and read it. $\begin(cases)y=\sqrt(x), x≥-1\\y=-x-2, x≤-1 \end(cases)$.
Solution. Let's construct two graphs of functions on the same coordinate plane, taking into account our conditions. For $x≥-1$ we build a graph of the cubic root, for $x≤-1$ we build a graph of a linear function.
1) D(y)=(-∞;+∞).
2) The function is neither even nor odd.
3) Decreases by (-∞;-1), increases by (-1;+∞).
4) Unlimited from above, limited from below.
5) There is no greatest value. The smallest value is minus one.
6) The function is continuous on the entire number line.
7) E(y)= (-1;+∞).

Problems to solve independently

1. Solve the equation $\sqrt(x)=2-x$.
2. Construct a graph of the function $y=\sqrt((x+1))+1$.
3.Plot a graph of the function and read it. $\begin(cases)y=\sqrt(x), x≥1\\y=(x-1)^2+1, x≤1 \end(cases)$.

Basic goals:

1) form an idea of the feasibility of a generalized study of the dependencies of real quantities using the example of quantities related by the relation y=

2) to develop the ability to construct a graph y= and its properties;

3) repeat and consolidate the techniques of oral and written calculations, squaring, extracting square roots.

Equipment, demonstration material: handouts.

1. Algorithm:

2. Sample for completing the task in groups:

3. Sample for self-test of independent work:

4. Card for the reflection stage:

1) I understood how to graph the function y=.

2) I can list its properties using a graph.

3) I did not make mistakes in independent work.

4) I made mistakes in my independent work (list these mistakes and indicate their reason).

During the classes

1. Self-determination for educational activities

Purpose of the stage:

1) include students in educational activities;

2) determine the content of the lesson: we continue to work with real numbers.

Organization of the educational process at stage 1:

– What did we study in the last lesson? (We studied the set of real numbers, operations with them, built an algorithm to describe the properties of a function, repeated functions studied in 7th grade).

– Today we will continue to work with a set of real numbers, a function.

2. Updating knowledge and recording difficulties in activities

Purpose of the stage:

1) update educational content that is necessary and sufficient for the perception of new material: function, independent variable, dependent variable, graphs

y = kx + m, y = kx, y =c, y =x 2, y = - x 2,

2) update mental operations necessary and sufficient for the perception of new material: comparison, analysis, generalization;

3) record all repeated concepts and algorithms in the form of diagrams and symbols;

4) record an individual difficulty in activity, demonstrating at a personally significant level the insufficiency of existing knowledge.

Organization of the educational process at stage 2:

1. Let's remember how you can set dependencies between quantities? (Using text, formula, table, graph)

2. What is a function called? (A relationship between two quantities, where each value of one variable corresponds to a single value of another variable y = f(x)).

What is the name of x? (Independent variable - argument)

What is the name of y? (Dependent variable).

3. In 7th grade did we study functions? (y = kx + m, y = kx, y =c, y =x 2, y = - x 2,).

Individual task:

What is the graph of the functions y = kx + m, y =x 2, y =?

3. Identifying the causes of difficulties and setting goals for activities

Purpose of the stage:

1) organize communicative interaction, during which the distinctive property a task that caused difficulty in learning activities;

2) agree on the purpose and topic of the lesson.

Organization of the educational process at stage 3:

-What's special about this task? (The dependence is given by the formula y = which we have not yet encountered.)

– What is the purpose of the lesson? (Get acquainted with the function y =, its properties and graph. Use the function in the table to determine the type of dependence, build a formula and graph.)

– Can you formulate the topic of the lesson? (Function y=, its properties and graph).

– Write the topic in your notebook.

4. Construction of a project for getting out of a difficulty

Purpose of the stage:

1) organize communicative interaction to build a new method of action that eliminates the cause of the identified difficulty;

2) fix new way actions in a symbolic, verbal form and using a standard.

Organization of the educational process at stage 4:

Work at this stage can be organized in groups, asking the groups to construct a graph y =, then analyze the results. Groups can also be asked to describe the properties of a given function using an algorithm.

5. Primary consolidation in external speech

The purpose of the stage: to record the studied educational content in external speech.

Organization of the educational process at stage 5:

Construct a graph of y= - and describe its properties.

Properties y= - .

1.Domain of definition of a function.

2. Range of values of the function.

3. y = 0, y> 0, y<0.

y =0 if x = 0.

y<0, если х(0;+)

4.Increasing, decreasing functions.

The function decreases as x.

Let's build a graph of y=.

Let's select its part on the segment. Note that we have = 1 for x = 1, and y max. =3 at x = 9.

Answer: at our name. = 1, y max. =3

6. Independent work with self-test according to the standard

The purpose of the stage: to test your ability to apply new educational content in standard conditions based on comparing your solution with a standard for self-test.

Organization of the educational process at stage 6:

Students complete the task independently, conduct a self-test against the standard, analyze, and correct errors.

Let's build a graph of y=.

Using a graph, find the smallest and largest values of the function on the segment.

7. Inclusion in the knowledge system and repetition

The purpose of the stage: to train the skills of using new content together with previously studied: 2) repeat the educational content that will be required in the next lessons.

Organization of the educational process at stage 7:

Solve the equation graphically: = x – 6.

One student is at the blackboard, the rest are in notebooks.

8. Reflection of activity

Purpose of the stage:

1) record new content learned in the lesson;

2) evaluate your own activities in the lesson;

3) thank classmates who helped get the result of the lesson;

4) record unresolved difficulties as directions for future educational activities;

5) discuss and write down your homework.

Organization of the educational process at stage 8:

- Guys, what was our goal today? (Study the function y=, its properties and graph).

– What knowledge helped us achieve our goal? (Ability to look for patterns, ability to read graphs.)

– Analyze your activities in class. (Cards with reflection)

Homework

paragraph 13 (before example 2) № 13.3, 13.4

Solve the equation graphically:

Construct a graph of the function and describe its properties.

Topic "Root of a degree" P"It is advisable to divide it into two lessons. In the first lesson, consider the cube root, compare its properties with the arithmetic square root and consider the graph of this Cube root function. Then in the second lesson, students will better understand the concept of a crown P-th degree. Comparing the two types of roots will help you avoid “typical” errors in the presence of values from negative expressions under the root sign.

View document contents
"Cubic root"

Lesson topic: Cube root

Zhikharev Sergey Alekseevich, mathematics teacher, MKOU “Pozhilinskaya Secondary School No. 13”

Lesson objectives:

introduce the concept of cube root;
develop skills in calculating cube roots;
repeat and generalize knowledge about the arithmetic square root;
continue preparing for the State Examination.

Checking the d.z.

One of the numbers below is marked on the coordinate line with a dot A. Enter this number.

What concept are the last three tasks related to?

What is the square root of a number? A ?

What is the arithmetic square root of a number? A ?

What values can the square root take?

Can a radical expression be a negative number?

Among these geometric bodies, name a cube

What properties does a cube have?

How to find the volume of a cube?

Find the volume of a cube if its sides are equal:

Let's solve the problem

The volume of the cube is 125 cm³. Find the side of the cube.

Let the edge of the cube be X cm, then the volume of the cube is X³ cm³. By condition X³ = 125.

Hence, X= 5 cm.

Number X= 5 is the root of the equation X³ = 125. This number is called cube root or third root from number 125.

Definition.

The third root of the number A this number is called b, the third power of which is equal to A .

Designation.

Another approach to introducing the concept of cube root

For a given cubic function value A, you can find the value of the argument of the cubic function at this point. It will be equal, since extracting a root is the inverse action of raising to a power.