Power functions with different exponents. Power function, its properties and graph

The power function is given by a formula of the form .

Consider the type of graphs of a power function and the properties of a power function depending on the value of the exponent.

Let's start with a power function with an integer exponent a. In this case, the form of graphs of power functions and the properties of functions depend on the even or odd exponent, as well as on its sign. Therefore, we first consider power functions for odd positive values ​​of the exponent a, then - for even positive, then - for odd negative exponents, and, finally, for even negative a.

The properties of power functions with fractional and irrational exponents (as well as the type of graphs of such power functions) depend on the value of the exponent a. We will consider them, firstly, a from zero to one, and secondly, at a large units, thirdly, with a from minus one to zero, fourthly, when a smaller minus one.

In conclusion of this subsection, for the sake of completeness, we describe a power function with zero exponent.

Power function with odd positive exponent.

Consider a power function with an odd positive exponent, that is, with a=1,3,5,….

The figure below shows graphs of power functions - black line, - blue line, - red line, - green line. At a=1 we have linear function y=x.

Properties of a power function with an odd positive exponent.

Power function with even positive exponent.

Consider a power function with an even positive exponent, that is, for a=2,4,6,….

As an example, let's take graphs of power functions - black line, - blue line, - red line. At a=2 we have a quadratic function whose graph is quadratic parabola.

Properties of a power function with an even positive exponent.

Power function with an odd negative exponent.

Look at the plots of the power function for odd negative values ​​of the exponent, that is, for a=-1,-3,-5,….

Knowledge basic elementary functions, their properties and graphs no less important than knowing the multiplication table. They are like a foundation, everything is based on them, everything is built from them, and everything comes down to them.

In this article, we list all the main elementary functions, give their graphs and give them without derivation and proofs. properties of basic elementary functions according to the scheme:

• behavior of the function on the boundaries of the domain of definition, vertical asymptotes (if necessary, see the article classification of breakpoints of a function);
• even and odd;
• convexity (convexity upwards) and concavity (convexity downwards) intervals, inflection points (if necessary, see the article function convexity, convexity direction, inflection points, convexity and inflection conditions);
• oblique and horizontal asymptotes;
• singular points of functions;
• special properties of some functions (for example, the smallest positive period for trigonometric functions).

If you are interested in or, then you can go to these sections of the theory.

Basic elementary functions are: constant function (constant), root of the nth degree, power function, exponential, logarithmic function, trigonometric and inverse trigonometric functions.

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Permanent function.

A constant function is given on the set of all real numbers by the formula , where C is some real number. The constant function assigns to each real value of the independent variable x the same value of the dependent variable y - the value С. A constant function is also called a constant.

The graph of a constant function is a straight line parallel to the x-axis and passing through a point with coordinates (0,C) . For example, let's show graphs of constant functions y=5 , y=-2 and , which in the figure below correspond to the black, red and blue lines, respectively.

Properties of a constant function.

• Domain of definition: the whole set of real numbers.
• The constant function is even.
• Range of values: set consisting of a single number C .
• A constant function is non-increasing and non-decreasing (that's why it is constant).
• It makes no sense to talk about the convexity and concavity of the constant.
• There is no asymptote.
• The function passes through the point (0,C) of the coordinate plane.

The root of the nth degree.

Consider the basic elementary function, which is given by the formula , where n is a natural number greater than one.

The root of the nth degree, n is an even number.

Let's start with the nth root function for even values ​​of the root exponent n .

For example, we give a picture with images of graphs of functions and , they correspond to black, red and blue lines.

The graphs of the functions of the root of an even degree have a similar form for other values ​​of the indicator.

Properties of the root of the nth degree for even n .

The root of the nth degree, n is an odd number.

The root function of the nth degree with an odd exponent of the root n is defined on the entire set of real numbers. For example, we present graphs of functions and , the black, red, and blue curves correspond to them.

For other odd values ​​of the root exponent, the graphs of the function will have a similar appearance.

Properties of the root of the nth degree for odd n .

Power function.

The power function is given by a formula of the form .

Consider the type of graphs of a power function and the properties of a power function depending on the value of the exponent.

Let's start with a power function with an integer exponent a . In this case, the form of graphs of power functions and the properties of functions depend on the even or odd exponent, as well as on its sign. Therefore, we first consider power functions for odd positive values ​​of the exponent a , then for even positive ones, then for odd negative exponents, and finally, for even negative a .

The properties of power functions with fractional and irrational exponents (as well as the type of graphs of such power functions) depend on the value of the exponent a. We will consider them, firstly, when a is from zero to one, secondly, when a is greater than one, thirdly, when a is from minus one to zero, and fourthly, when a is less than minus one.

In conclusion of this subsection, for the sake of completeness, we describe a power function with zero exponent.

Power function with odd positive exponent.

Consider a power function with an odd positive exponent, that is, with a=1,3,5,… .

The figure below shows graphs of power functions - black line, - blue line, - red line, - green line. For a=1 we have linear function y=x .

Properties of a power function with an odd positive exponent.

Power function with even positive exponent.

Consider a power function with an even positive exponent, that is, for a=2,4,6,… .

As an example, let's take graphs of power functions - black line, - blue line, - red line. For a=2 we have a quadratic function whose graph is quadratic parabola.

Properties of a power function with an even positive exponent.

Power function with an odd negative exponent.

Look at the graphs of the exponential function for odd negative values ​​​​of the exponent, that is, for a \u003d -1, -3, -5, ....

The figure shows graphs of exponential functions as examples - black line, - blue line, - red line, - green line. For a=-1 we have inverse proportionality, whose graph is hyperbola.

Properties of a power function with an odd negative exponent.

Power function with an even negative exponent.

Let's move on to the power function at a=-2,-4,-6,….

The figure shows graphs of power functions - black line, - blue line, - red line.

Properties of a power function with an even negative exponent.

A power function with a rational or irrational exponent whose value is greater than zero and less than one.

Note! If a is a positive fraction with an odd denominator, then some authors consider the interval to be the domain of the power function. At the same time, it is stipulated that the exponent a is an irreducible fraction. Now the authors of many textbooks on algebra and the beginnings of analysis DO NOT DEFINE power functions with an exponent in the form of a fraction with an odd denominator for negative values ​​of the argument. We will adhere to just such a view, that is, we will consider the domains of power functions with fractional positive exponents to be the set . We encourage students to get your teacher's perspective on this subtle point to avoid disagreement.

Consider a power function with rational or irrational exponent a , and .

We present graphs of power functions for a=11/12 (black line), a=5/7 (red line), (blue line), a=2/5 (green line).

A power function with a non-integer rational or irrational exponent greater than one.

Consider a power function with a non-integer rational or irrational exponent a , and .

Let us present the graphs of the power functions given by the formulas (black, red, blue and green lines respectively).

For other values ​​of the exponent a , the graphs of the function will have a similar look.

Power function properties for .

A power function with a real exponent that is greater than minus one and less than zero.

Note! If a is a negative fraction with an odd denominator, then some authors consider the interval . At the same time, it is stipulated that the exponent a is an irreducible fraction. Now the authors of many textbooks on algebra and the beginnings of analysis DO NOT DEFINE power functions with an exponent in the form of a fraction with an odd denominator for negative values ​​of the argument. We will adhere to just such a view, that is, we will consider the domains of power functions with fractional negative exponents to be the set, respectively. We encourage students to get your teacher's perspective on this subtle point to avoid disagreement.

We pass to the power function , where .

In order to have a good idea of ​​the type of graphs of power functions for , we give examples of graphs of functions (black, red, blue, and green curves, respectively).

Properties of a power function with exponent a , .

A power function with a non-integer real exponent that is less than minus one.

Let us give examples of graphs of power functions for , they are depicted in black, red, blue and green lines, respectively.

Properties of a power function with a non-integer negative exponent less than minus one.

When a=0 and we have a function - this is a straight line from which the point (0; 1) is excluded (the expression 0 0 was agreed not to attach any importance).

Exponential function.

One of the basic elementary functions is the exponential function.

Graph of the exponential function, where and takes a different form depending on the value of the base a. Let's figure it out.

First, consider the case when the base of the exponential function takes a value from zero to one, that is, .

For example, we present the graphs of the exponential function for a = 1/2 - the blue line, a = 5/6 - the red line. The graphs of the exponential function have a similar appearance for other values ​​of the base from the interval .

Properties of an exponential function with a base less than one.

We turn to the case when the base of the exponential function is greater than one, that is, .

As an illustration, we present graphs of exponential functions - the blue line and - the red line. For other values ​​​​of the base, greater than one, the graphs of the exponential function will have a similar appearance.

Properties of an exponential function with a base greater than one.

Logarithmic function.

The next basic elementary function is the logarithmic function , where , . The logarithmic function is defined only for positive values ​​of the argument, that is, for .

The graph of the logarithmic function takes on a different form depending on the value of the base a.

Let's start with the case when .

For example, we present the graphs of the logarithmic function for a = 1/2 - the blue line, a = 5/6 - the red line. For other values ​​​​of the base, not exceeding one, the graphs of the logarithmic function will have a similar appearance.

Properties of a logarithmic function with a base less than one.

Let's move on to the case when the base of the logarithmic function is greater than one ().

Let's show graphs of logarithmic functions - blue line, - red line. For other values ​​​​of the base, greater than one, the graphs of the logarithmic function will have a similar appearance.

Properties of a logarithmic function with a base greater than one.

Trigonometric functions, their properties and graphs.

All trigonometric functions (sine, cosine, tangent and cotangent) are basic elementary functions. Now we will consider their graphs and list their properties.

Trigonometric functions have the concept periodicity(recurrence of function values ​​for different values ​​of the argument that differ from each other by the value of the period , where T is the period), therefore, an item has been added to the list of properties of trigonometric functions "smallest positive period". Also, for each trigonometric function, we will indicate the values ​​of the argument at which the corresponding function vanishes.

Now let's deal with all the trigonometric functions in order.

The sine function y = sin(x) .

Let's draw a graph of the sine function, it is called a "sinusoid".

Properties of the sine function y = sinx .

Cosine function y = cos(x) .

The graph of the cosine function (it is called "cosine") looks like this:

Cosine function properties y = cosx .

Tangent function y = tg(x) .

The graph of the tangent function (it is called the "tangentoid") looks like:

Function properties tangent y = tgx .

Cotangent function y = ctg(x) .

Let's draw a graph of the cotangent function (it's called a "cotangentoid"):

Cotangent function properties y = ctgx .

Inverse trigonometric functions, their properties and graphs.

The inverse trigonometric functions (arcsine, arccosine, arctangent and arccotangent) are the basic elementary functions. Often, because of the prefix "arc", inverse trigonometric functions are called arc functions. Now we will consider their graphs and list their properties.

Arcsine function y = arcsin(x) .

Let's plot the arcsine function:

Bibliography.

• Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. Algebra and the Beginnings of Analysis: Proc. for 10-11 cells. educational institutions.
• Vygodsky M.Ya. Handbook of elementary mathematics.
• Novoselov S.I. Algebra and elementary functions.
• Tumanov S.I. Elementary Algebra. A guide for self-education.

Are you familiar with the features y=x, y=x 2 , y=x 3 , y=1/x etc. All these functions are special cases of the power function, i.e., the function y=x p, where p is a given real number. The properties and graph of a power function essentially depend on the properties of a power with a real exponent, and in particular on the values ​​for which x and p makes sense x p. Let us proceed to a similar consideration of various cases depending on the exponent p.

Index p=2n is an even natural number.

In this case, the power function y=x 2n, where n is a natural number, has the following

properties:

the domain of definition is all real numbers, i.e., the set R;

set of values ​​- non-negative numbers, i.e. y is greater than or equal to 0;

function y=x 2n even, because x 2n =(-x) 2n

the function is decreasing on the interval x<0 and increasing on the interval x>0.

Function Graph y=x 2n has the same form as, for example, the graph of a function y=x 4 .

2. Indicator p=2n-1- odd natural number In this case, the power function y=x 2n-1, where is a natural number, has the following properties:

domain of definition - set R;

set of values ​​- set R;

function y=x 2n-1 odd because (- x) 2n-1 =x 2n-1 ;

the function is increasing on the entire real axis.

Function Graph y=x2n-1 has the same form as, for example, the graph of the function y=x3.

3.Indicator p=-2n, where n- natural number.

In this case, the power function y=x -2n =1/x 2n has the following properties:

set of values ​​- positive numbers y>0;

function y =1/x 2n even, because 1/(-x) 2n =1/x 2n ;

the function is increasing on the interval x<0 и убывающей на промежутке x>0.

Graph of the function y =1/x 2n has the same form as, for example, the graph of the function y =1/x 2 .

4.Indicator p=-(2n-1), where n- natural number. In this case, the power function y=x -(2n-1) has the following properties:

domain of definition - set R, except for x=0;

set of values ​​- set R, except for y=0;

function y=x -(2n-1) odd because (- x) -(2n-1) =-x -(2n-1) ;

the function is decreasing on the intervals x<0 and x>0.

Function Graph y=x -(2n-1) has the same form as, for example, the graph of the function y=1/x 3 .

1. Inverse trigonometric functions, their properties and graphs.

Inverse trigonometric functions, their properties and graphs.Inverse trigonometric functions (circular functions, arcfunctions) are mathematical functions that are inverse to trigonometric functions.

1. arcsin function

Function Graph .

arcsine numbers m is called such an angle x, for which

The function is continuous and bounded on its entire real line. Function is strictly increasing.

1. [Edit] Properties of the arcsin function

1. [Edit] Getting the arcsin function

Given a function Throughout its domains she happens to be piecewise monotonic, and hence the inverse correspondence is not a function. Therefore, we consider the interval on which it strictly increases and takes all values ranges- . Since for a function on the interval, each value of the argument corresponds to a single value of the function, then on this segment there exists inverse function whose graph is symmetrical to the graph of a function on a segment with respect to a straight line

Lecture: Power function with a natural exponent, its graph

We are constantly dealing with functions in which the argument has some power:
y \u003d x 1, y \u003d x 2, y \u003d x 3, y \u003d x -1, etc.

Graphs of Power Functions

So, now we will consider several possible cases of a power function.

1) y = x 2 n .

This means that now we will consider functions in which the exponent is an even number.

Feature Feature:

1. All real numbers are accepted as the range.

2. The function can take all positive values ​​and the number zero.

3. The function is even because it does not depend on the sign of the argument, but only on its modulus.

4. For a positive argument, the function is increasing, and for a negative one, it is decreasing.

The graphs of these functions resemble a parabola. For example, below is a graph of the function y \u003d x 4.

2) The function has an odd exponent: y \u003d x 2 n +1.

1. The domain of the function is the entire set of real numbers.

2. Function range - can take the form of any real number.

3. This function is odd.

4. Monotonically increases over the entire interval of considering the function.

5. The graph of all power functions with an odd exponent is identical to the function y \u003d x 3.

3) The function has an even negative natural exponent: y \u003d x -2 n.

We all know that a negative exponent allows you to drop the exponent into the denominator and change the sign of the exponent, that is, you get the form y \u003d 1 / x 2 n.

1. The argument of this function can take any value except zero, since the variable is in the denominator.

2. Since the exponent is an even number, the function cannot take negative values. And since the argument cannot be equal to zero, then the value of the function equal to zero should also be excluded. This means that the function can only take positive values.

3. This function is even.

4. If the argument is negative, the function is monotonically increasing, and if it is positive, it is decreasing.

View of the graph of the function y \u003d x -2:

4) Function with negative odd exponent y \u003d x - (2 n + 1) .

1. This function exists for all values ​​of the argument, except for the number zero.

2. The function accepts all real values, except for the number zero.

3. This function is odd.

4. Decreases on the two considered intervals.

Consider an example of a graph of a function with a negative odd exponent using the example y \u003d x -3.

On the domain of the power function y = x p, the following formulas hold:
; ;
;
; ;
; ;
; .

Properties of power functions and their graphs

Power function with exponent equal to zero, p = 0

If the exponent of the power function y = x p is equal to zero, p = 0 , then the power function is defined for all x ≠ 0 and is constant, equal to one:
y \u003d x p \u003d x 0 \u003d 1, x ≠ 0.

Power function with natural odd exponent, p = n = 1, 3, 5, ...

Consider a power function y = x p = x n with natural odd exponent n = 1, 3, 5, ... . Such an indicator can also be written as: n = 2k + 1, where k = 0, 1, 2, 3, ... is a non-negative integer. Below are the properties and graphs of such functions.

Graph of a power function y = x n with a natural odd exponent for various values ​​of the exponent n = 1, 3, 5, ... .

Domain: -∞ < x < ∞
Multiple values: -∞ < y < ∞
Parity: odd, y(-x) = - y(x)
Monotone: increases monotonically
Extremes: No
Convex:
at -∞< x < 0 выпукла вверх
at 0< x < ∞ выпукла вниз
Breakpoints: x=0, y=0
x=0, y=0
Limits:
;
Private values:
at x = -1,
y(-1) = (-1) n ≡ (-1) 2k+1 = -1
for x = 0, y(0) = 0 n = 0
for x = 1, y(1) = 1 n = 1
Reverse function:
for n = 1 , the function is inverse to itself: x = y
for n ≠ 1, the inverse function is a root of degree n:

Power function with natural even exponent, p = n = 2, 4, 6, ...

Consider a power function y = x p = x n with natural even exponent n = 2, 4, 6, ... . Such an indicator can also be written as: n = 2k, where k = 1, 2, 3, ... is a natural number. The properties and graphs of such functions are given below.

Graph of a power function y = x n with a natural even exponent for various values ​​of the exponent n = 2, 4, 6, ... .

Domain: -∞ < x < ∞
Multiple values: 0 ≤ y< ∞
Parity: even, y(-x) = y(x)
Monotone:
for x ≤ 0 monotonically decreases
for x ≥ 0 monotonically increases
Extremes: minimum, x=0, y=0
Convex: convex down
Breakpoints: No
Intersection points with coordinate axes: x=0, y=0
Limits:
;
Private values:
for x = -1, y(-1) = (-1) n ≡ (-1) 2k = 1
for x = 0, y(0) = 0 n = 0
for x = 1, y(1) = 1 n = 1
Reverse function:
for n = 2, square root:
for n ≠ 2, root of degree n:

Power function with integer negative exponent, p = n = -1, -2, -3, ...

Consider a power function y = x p = x n with a negative integer exponent n = -1, -2, -3, ... . If we put n = -k, where k = 1, 2, 3, ... is a natural number, then it can be represented as:

Graph of a power function y = x n with a negative integer exponent for various values ​​of the exponent n = -1, -2, -3, ... .

Odd exponent, n = -1, -3, -5, ...

Below are the properties of the function y = x n with an odd negative exponent n = -1, -3, -5, ... .

Domain: x ≠ 0
Multiple values: y ≠ 0
Parity: odd, y(-x) = - y(x)
Monotone: decreases monotonically
Extremes: No
Convex:
at x< 0 : выпукла вверх
for x > 0 : convex down
Breakpoints: No
Intersection points with coordinate axes: No
Sign:
at x< 0, y < 0
for x > 0, y > 0
Limits:
; ; ;
Private values:
for x = 1, y(1) = 1 n = 1
Reverse function:
for n = -1,
for n< -2 ,

Even exponent, n = -2, -4, -6, ...

Below are the properties of the function y = x n with an even negative exponent n = -2, -4, -6, ... .

Domain: x ≠ 0
Multiple values: y > 0
Parity: even, y(-x) = y(x)
Monotone:
at x< 0 : монотонно возрастает
for x > 0 : monotonically decreasing
Extremes: No
Convex: convex down
Breakpoints: No
Intersection points with coordinate axes: No
Sign: y > 0
Limits:
; ; ;
Private values:
for x = 1, y(1) = 1 n = 1
Reverse function:
for n = -2,
for n< -2 ,

Power function with rational (fractional) exponent

Consider a power function y = x p with a rational (fractional) exponent , where n is an integer, m > 1 is a natural number. Moreover, n, m do not have common divisors.

The denominator of the fractional indicator is odd

Let the denominator of the fractional exponent be odd: m = 3, 5, 7, ... . In this case, the power function x p is defined for both positive and negative x values. Consider the properties of such power functions when the exponent p is within certain limits.

p is negative, p< 0

Let the rational exponent (with odd denominator m = 3, 5, 7, ... ) be less than zero: .

Graphs of exponential functions with a rational negative exponent for various values ​​of the exponent , where m = 3, 5, 7, ... is odd.

Odd numerator, n = -1, -3, -5, ...

Here are the properties of the power function y = x p with a rational negative exponent , where n = -1, -3, -5, ... is an odd negative integer, m = 3, 5, 7 ... is an odd natural number.

Domain: x ≠ 0
Multiple values: y ≠ 0
Parity: odd, y(-x) = - y(x)
Monotone: decreases monotonically
Extremes: No
Convex:
at x< 0 : выпукла вверх
for x > 0 : convex down
Breakpoints: No
Intersection points with coordinate axes: No
Sign:
at x< 0, y < 0
for x > 0, y > 0
Limits:
; ; ;
Private values:
for x = -1, y(-1) = (-1) n = -1
for x = 1, y(1) = 1 n = 1
Reverse function:

Even numerator, n = -2, -4, -6, ...

Properties of a power function y = x p with a rational negative exponent, where n = -2, -4, -6, ... is an even negative integer, m = 3, 5, 7 ... is an odd natural number.

Domain: x ≠ 0
Multiple values: y > 0
Parity: even, y(-x) = y(x)
Monotone:
at x< 0 : монотонно возрастает
for x > 0 : monotonically decreasing
Extremes: No
Convex: convex down
Breakpoints: No
Intersection points with coordinate axes: No
Sign: y > 0
Limits:
; ; ;
Private values:
for x = -1, y(-1) = (-1) n = 1
for x = 1, y(1) = 1 n = 1
Reverse function:

The p-value is positive, less than one, 0< p < 1

Graph of a power function with a rational exponent (0< p < 1 ) при различных значениях показателя степени , где m = 3, 5, 7, ... - нечетное.

Odd numerator, n = 1, 3, 5, ...

< p < 1 , где n = 1, 3, 5, ... - нечетное натуральное, m = 3, 5, 7 ... - нечетное натуральное.

Domain: -∞ < x < +∞
Multiple values: -∞ < y < +∞
Parity: odd, y(-x) = - y(x)
Monotone: increases monotonically
Extremes: No
Convex:
at x< 0 : выпукла вниз
for x > 0 : convex up
Breakpoints: x=0, y=0
Intersection points with coordinate axes: x=0, y=0
Sign:
at x< 0, y < 0
for x > 0, y > 0
Limits:
;
Private values:
for x = -1, y(-1) = -1
for x = 0, y(0) = 0
for x = 1, y(1) = 1
Reverse function:

Even numerator, n = 2, 4, 6, ...

The properties of the power function y = x p with a rational exponent , being within 0 are presented.< p < 1 , где n = 2, 4, 6, ... - четное натуральное, m = 3, 5, 7 ... - нечетное натуральное.

Domain: -∞ < x < +∞
Multiple values: 0 ≤ y< +∞
Parity: even, y(-x) = y(x)
Monotone:
at x< 0 : монотонно убывает
for x > 0 : monotonically increasing
Extremes: minimum at x = 0, y = 0
Convex: convex upward at x ≠ 0
Breakpoints: No
Intersection points with coordinate axes: x=0, y=0
Sign: for x ≠ 0, y > 0
Limits:
;
Private values:
for x = -1, y(-1) = 1
for x = 0, y(0) = 0
for x = 1, y(1) = 1
Reverse function:

The exponent p is greater than one, p > 1

Graph of a power function with a rational exponent (p > 1 ) for various values ​​of the exponent , where m = 3, 5, 7, ... is odd.

Odd numerator, n = 5, 7, 9, ...

Properties of a power function y = x p with a rational exponent greater than one: . Where n = 5, 7, 9, ... is an odd natural number, m = 3, 5, 7 ... is an odd natural number.

Domain: -∞ < x < ∞
Multiple values: -∞ < y < ∞
Parity: odd, y(-x) = - y(x)
Monotone: increases monotonically
Extremes: No
Convex:
at -∞< x < 0 выпукла вверх
at 0< x < ∞ выпукла вниз
Breakpoints: x=0, y=0
Intersection points with coordinate axes: x=0, y=0
Limits:
;
Private values:
for x = -1, y(-1) = -1
for x = 0, y(0) = 0
for x = 1, y(1) = 1
Reverse function:

Even numerator, n = 4, 6, 8, ...

Properties of a power function y = x p with a rational exponent greater than one: . Where n = 4, 6, 8, ... is an even natural number, m = 3, 5, 7 ... is an odd natural number.

Domain: -∞ < x < ∞
Multiple values: 0 ≤ y< ∞
Parity: even, y(-x) = y(x)
Monotone:
at x< 0 монотонно убывает
for x > 0 monotonically increases
Extremes: minimum at x = 0, y = 0
Convex: convex down
Breakpoints: No
Intersection points with coordinate axes: x=0, y=0
Limits:
;
Private values:
for x = -1, y(-1) = 1
for x = 0, y(0) = 0
for x = 1, y(1) = 1
Reverse function:

The denominator of the fractional indicator is even

Let the denominator of the fractional exponent be even: m = 2, 4, 6, ... . In this case, the power function x p is not defined for negative values ​​of the argument. Its properties coincide with those of a power function with an irrational exponent (see the next section).

Power function with irrational exponent

Consider a power function y = x p with an irrational exponent p . The properties of such functions differ from those considered above in that they are not defined for negative values ​​of the x argument. For positive values ​​of the argument, the properties depend only on the value of the exponent p and do not depend on whether p is integer, rational, or irrational.

Power function with negative p< 0

Domain: x > 0
Multiple values: y > 0
Monotone: decreases monotonically
Convex: convex down
Breakpoints: No
Intersection points with coordinate axes: No
Limits: ;
private value: For x = 1, y(1) = 1 p = 1

Power function with positive exponent p > 0

The indicator is less than one 0< p < 1

Domain: x ≥ 0
Multiple values: y ≥ 0
Monotone: increases monotonically
Convex: convex up
Breakpoints: No
Intersection points with coordinate axes: x=0, y=0
Limits:
Private values: For x = 0, y(0) = 0 p = 0 .
For x = 1, y(1) = 1 p = 1

The indicator is greater than one p > 1

Domain: x ≥ 0
Multiple values: y ≥ 0
Monotone: increases monotonically
Convex: convex down
Breakpoints: No
Intersection points with coordinate axes: x=0, y=0
Limits:
Private values: For x = 0, y(0) = 0 p = 0 .
For x = 1, y(1) = 1 p = 1

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Higher Educational Institutions, Lan, 2009.