Main elementary functions, their inherent properties and corresponding graphs are one of the basics of mathematical knowledge, similar in importance to the multiplication table. Elementary functions are the basis, the support for the study of all theoretical issues.

The article below provides key material on the topic of basic elementary functions. We will introduce terms, give them definitions; Let us study in detail each type of elementary functions and analyze their properties.

The following types of basic elementary functions are distinguished:

Definition 1

• constant function (constant);
• root of the nth degree;
• power function;
• exponential function;
• logarithmic function;
• trigonometric functions;
• fraternal trigonometric functions.

A constant function is defined by the formula: y = C (C is some real number) and also has a name: constant. This function determines whether any real value of the independent variable x corresponds to the same value of the variable y – the value C .

The graph of a constant is a straight line that is parallel to the x-axis and passes through a point having coordinates (0, C). For clarity, we present graphs of constant functions y = 5 , y = - 2 , y = 3 , y = 3 (marked in black, red and blue in the drawing, respectively).

Definition 2

This elementary function is defined by the formula y = x n (n - natural number more than one).

Let's consider two variations of the function.

1. Root of the nth degree, n is an even number

For clarity, we indicate the drawing, which shows the graphs of such functions: y = x , y = x 4 and y = x 8 . These functions are color-coded: black, red and blue, respectively.

A similar view of the graphs of the function of an even degree for other values ​​​​of the indicator.

Definition 3

Properties of the function root of the nth degree, n is an even number

• the domain of definition is the set of all non-negative real numbers [ 0 , + ∞) ;
• when x = 0 , the function y = x n has a value equal to zero;
• given function - function general view(is neither even nor odd);
• range: [ 0 , + ∞) ;
• this function y = x n with even exponents of the root increases over the entire domain of definition;
• the function has a convexity with upward direction over the entire domain of definition;
• there are no inflection points;
• there are no asymptotes;
• the graph of the function for even n passes through the points (0 ; 0) and (1 ; 1) .

1. Root of the nth degree, n is an odd number

Such a function is defined on the entire set of real numbers. For clarity, consider the graphs of functions y = x 3 , y = x 5 and x 9 . In the drawing, they are indicated by colors: black, red and blue colors of the curves, respectively.

Other odd values ​​of the exponent of the root of the function y = x n will give a graph of a similar form.

Definition 4

Properties of the function root of the nth degree, n is an odd number

• the domain of definition is the set of all real numbers;
• this function is odd;
• the range of values ​​is the set of all real numbers;
• the function y = x n with odd exponents of the root increases over the entire domain of definition;
• the function has concavity on the interval (- ∞ ; 0 ] and convexity on the interval [ 0 , + ∞) ;
• the inflection point has coordinates (0 ; 0) ;
• there are no asymptotes;
• the graph of the function for odd n passes through the points (- 1 ; - 1) , (0 ; 0) and (1 ; 1) .

Power function

Definition 5

The power function is defined by the formula y = x a .

The type of graphs and properties of the function depend on the value of the exponent.

• when a power function has an integer exponent a, then the form of the graph of the power function and its properties depend on whether the exponent is even or odd, and also what sign the exponent has. Let us consider all these special cases in more detail below;
• the exponent can be fractional or irrational - depending on this, the type of graphs and the properties of the function also vary. We will analyze special cases by setting several conditions: 0< a < 1 ; a > 1 ; - 1 < a < 0 и a < - 1 ;
• a power function can have a zero exponent, we will also analyze this case in more detail below.

Let's analyze the power function y = x a when a is odd positive number, for example, a = 1 , 3 , 5 ...

For clarity, we indicate the graphs of such power functions: y = x (black color of the graph), y = x 3 (blue color of the chart), y = x 5 (red color of the graph), y = x 7 (green graph). When a = 1 , we get a linear function y = x .

Definition 6

Properties of a power function when the exponent is an odd positive

• the function is increasing for x ∈ (- ∞ ; + ∞) ;
• the function is convex for x ∈ (- ∞ ; 0 ] and concave for x ∈ [ 0 ; + ∞) (excluding the linear function);
• the inflection point has coordinates (0 ; 0) (excluding the linear function);
• there are no asymptotes;
• function passing points: (- 1 ; - 1) , (0 ; 0) , (1 ; 1) .

Let's analyze the power function y = x a when a is an even positive number, for example, a = 2 , 4 , 6 ...

For clarity, we indicate the graphs of such power functions: y \u003d x 2 (black color of the graph), y = x 4 (blue color of the graph), y = x 8 (red color of the graph). When a = 2, we get a quadratic function whose graph is a quadratic parabola.

Definition 7

Properties of a power function when the exponent is even positive:

• domain of definition: x ∈ (- ∞ ; + ∞) ;
• decreasing for x ∈ (- ∞ ; 0 ] ;
• the function is concave for x ∈ (- ∞ ; + ∞) ;
• there are no inflection points;
• there are no asymptotes;
• function passing points: (- 1 ; 1) , (0 ; 0) , (1 ; 1) .

The figure below shows examples of exponential function graphs y = x a when a is odd a negative number: y = x - 9 (black color of the chart); y = x - 5 (blue color of the graph); y = x - 3 (red color of the chart); y = x - 1 (green graph). When a \u003d - 1, we get an inverse proportionality, the graph of which is a hyperbola.

Definition 8

Power function properties when the exponent is odd negative:

When x \u003d 0, we get a discontinuity of the second kind, since lim x → 0 - 0 x a \u003d - ∞, lim x → 0 + 0 x a \u003d + ∞ for a \u003d - 1, - 3, - 5, .... Thus, the straight line x = 0 is a vertical asymptote;

• range: y ∈ (- ∞ ; 0) ∪ (0 ; + ∞) ;
• the function is odd because y (- x) = - y (x) ;
• the function is decreasing for x ∈ - ∞ ; 0 ∪ (0 ; + ∞) ;
• the function is convex for x ∈ (- ∞ ; 0) and concave for x ∈ (0 ; + ∞) ;
• there are no inflection points;

k = lim x → ∞ x a x = 0 , b = lim x → ∞ (x a - k x) = 0 ⇒ y = k x + b = 0 when a = - 1 , - 3 , - 5 , . . . .

• function passing points: (- 1 ; - 1) , (1 ; 1) .

The figure below shows examples of power function graphs y = x a when a is an even negative number: y = x - 8 (chart in black); y = x - 4 (blue color of the graph); y = x - 2 (red color of the graph).

Definition 9

Power function properties when the exponent is even negative:

• domain of definition: x ∈ (- ∞ ; 0) ∪ (0 ; + ∞) ;

When x \u003d 0, we get a discontinuity of the second kind, since lim x → 0 - 0 x a \u003d + ∞, lim x → 0 + 0 x a \u003d + ∞ for a \u003d - 2, - 4, - 6, .... Thus, the straight line x = 0 is a vertical asymptote;

• the function is even because y (- x) = y (x) ;
• the function is increasing for x ∈ (- ∞ ; 0) and decreasing for x ∈ 0 ; +∞ ;
• the function is concave for x ∈ (- ∞ ; 0) ∪ (0 ; + ∞) ;
• there are no inflection points;
• the horizontal asymptote is a straight line y = 0 because:

k = lim x → ∞ x a x = 0 , b = lim x → ∞ (x a - k x) = 0 ⇒ y = k x + b = 0 when a = - 2 , - 4 , - 6 , . . . .

• function passing points: (- 1 ; 1) , (1 ; 1) .

From the very beginning, pay attention to the following aspect: in the case when a is a positive fraction with an odd denominator, some authors take the interval - ∞ as the domain of definition of this power function; + ∞ , stipulating that the exponent a is an irreducible fraction. At present, the authors of many educational publications according to algebra and the beginnings of analysis, power functions are NOT DEFINED, where the exponent is a fraction with an odd denominator for negative values ​​of the argument. Further, we will adhere to just such a position: we take the set [ 0 ; +∞) . Recommendation for students: find out the teacher's point of view at this point in order to avoid disagreements.

So let's take a look at the power function y = x a when the exponent is a rational or irrational number provided that 0< a < 1 .

Let us illustrate with graphs the power functions y = x a when a = 11 12 (chart in black); a = 5 7 (red color of the graph); a = 1 3 (blue color of the chart); a = 2 5 (green color of the graph).

Other values ​​of the exponent a (assuming 0< a < 1) дадут аналогичный вид графика.

Definition 10

Power function properties at 0< a < 1:

• range: y ∈ [ 0 ; +∞) ;
• the function is increasing for x ∈ [ 0 ; +∞) ;
• the function has convexity for x ∈ (0 ; + ∞) ;
• there are no inflection points;
• there are no asymptotes;

Let's analyze the power function y = x a when the exponent is a non-integer rational or irrational number provided that a > 1 .

We illustrate the graphs of the power function y = x a under given conditions on the example of such functions: y = x 5 4 , y = x 4 3 , y = x 7 3 , y = x 3 π (black, red, blue, green color of graphs respectively).

Other values ​​of the exponent a under the condition a > 1 will give a similar view of the graph.

Definition 11

Power function properties for a > 1:

• domain of definition: x ∈ [ 0 ; +∞) ;
• range: y ∈ [ 0 ; +∞) ;
• this function is a function of general form (it is neither odd nor even);
• the function is increasing for x ∈ [ 0 ; +∞) ;
• the function is concave for x ∈ (0 ; + ∞) (when 1< a < 2) и выпуклость при x ∈ [ 0 ; + ∞) (когда a > 2);
• there are no inflection points;
• there are no asymptotes;
• function passing points: (0 ; 0) , (1 ; 1) .

We draw your attention! When a is a negative fraction with an odd denominator, in the works of some authors there is a view that the domain of definition in this case– interval - ∞ ; 0 ∪ (0 ; + ∞) with the proviso that the exponent a is an irreducible fraction. At the moment the authors teaching materials according to algebra and the beginnings of analysis, power functions with an exponent in the form of a fraction with an odd denominator with negative values ​​of the argument are NOT DEFINED. Further, we adhere to just such a view: we take the set (0 ; + ∞) as the domain of power functions with fractional negative exponents. Suggestion for students: Clarify your teacher's vision at this point to avoid disagreement.

We continue the topic and analyze the power function y = x a provided: - 1< a < 0 .

Here is a drawing of graphs of the following functions: y = x - 5 6 , y = x - 2 3 , y = x - 1 2 2 , y = x - 1 7 (black, red, blue, green lines, respectively).

Definition 12

Power function properties at - 1< a < 0:

lim x → 0 + 0 x a = + ∞ when - 1< a < 0 , т.е. х = 0 – вертикальная асимптота;

• range: y ∈ 0 ; +∞ ;
• this function is a function of general form (it is neither odd nor even);
• there are no inflection points;

The drawing below shows graphs of power functions y = x - 5 4 , y = x - 5 3 , y = x - 6 , y = x - 24 7 (black, red, blue, green colors curves, respectively).

Definition 13

Power function properties for a< - 1:

• domain of definition: x ∈ 0 ; +∞ ;

lim x → 0 + 0 x a = + ∞ when a< - 1 , т.е. х = 0 – вертикальная асимптота;

• range: y ∈ (0 ; + ∞) ;
• this function is a function of general form (it is neither odd nor even);
• the function is decreasing for x ∈ 0; +∞ ;
• the function is concave for x ∈ 0; +∞ ;
• there are no inflection points;
• horizontal asymptote - straight line y = 0 ;
• function passing point: (1 ; 1) .

When a \u003d 0 and x ≠ 0, we get the function y \u003d x 0 \u003d 1, which determines the line from which the point (0; 1) is excluded (we agreed that the expression 0 0 will not be given any value).

The exponential function has the form y = a x , where a > 0 and a ≠ 1 , and the graph of this function looks different based on the value of the base a . Let's consider special cases.

First, let's analyze the situation when the base of the exponential function has a value from zero to one (0< a < 1) . An illustrative example is the graphs of functions for a = 1 2 (blue color of the curve) and a = 5 6 (red color of the curve).

The graphs of the exponential function will have a similar form for other values ​​​​of the base, provided that 0< a < 1 .

Definition 14

Properties of an exponential function when the base is less than one:

• range: y ∈ (0 ; + ∞) ;
• this function is a function of general form (it is neither odd nor even);
• an exponential function whose base is less than one is decreasing over the entire domain of definition;
• there are no inflection points;
• the horizontal asymptote is the straight line y = 0 with the variable x tending to + ∞ ;

Now consider the case when the base of the exponential function is greater than one (a > 1).

Let's illustrate this special case with the graph of exponential functions y = 3 2 x (blue color of the curve) and y = e x (red color of the graph).

Other values ​​​​of the base, greater than one, will give a similar view of the graph of the exponential function.

Definition 15

Properties of the exponential function when the base is greater than one:

• the domain of definition is the entire set of real numbers;
• range: y ∈ (0 ; + ∞) ;
• this function is a function of general form (it is neither odd nor even);
• an exponential function whose base is greater than one is increasing for x ∈ - ∞ ; +∞ ;
• the function is concave for x ∈ - ∞ ; +∞ ;
• there are no inflection points;
• horizontal asymptote - straight line y = 0 with variable x tending to - ∞ ;
• function passing point: (0 ; 1) .

The logarithmic function has the form y = log a (x) , where a > 0 , a ≠ 1 .

Such a function is defined only for positive values ​​of the argument: for x ∈ 0 ; +∞ .

The plot of the logarithmic function has different kind, based on the value of the base a.

Consider first the situation when 0< a < 1 . Продемонстрируем этот частный случай графиком логарифмической функции при a = 1 2 (синий цвет кривой) и а = 5 6 (красный цвет кривой).

Other values ​​of the base, not greater than one, will give a similar view of the graph.

Definition 16

Properties of a logarithmic function when the base is less than one:

• domain of definition: x ∈ 0 ; +∞ . As x tends to zero from the right, the values ​​of the function tend to + ∞;
• range: y ∈ - ∞ ; +∞ ;
• this function is a function of general form (it is neither odd nor even);
• logarithmic
• the function is concave for x ∈ 0; +∞ ;
• there are no inflection points;
• there are no asymptotes;

Now let's analyze a special case when the base of the logarithmic function is greater than one: a > 1 . In the drawing below, there are graphs of logarithmic functions y = log 3 2 x and y = ln x (blue and red colors of the graphs, respectively).

Other values ​​of the base greater than one will give a similar view of the graph.

Definition 17

Properties of a logarithmic function when the base is greater than one:

• domain of definition: x ∈ 0 ; +∞ . As x tends to zero from the right, the values ​​of the function tend to - ∞;
• range: y ∈ - ∞ ; + ∞ (the whole set of real numbers);
• this function is a function of general form (it is neither odd nor even);
• the logarithmic function is increasing for x ∈ 0; +∞ ;
• the function has convexity for x ∈ 0; +∞ ;
• there are no inflection points;
• there are no asymptotes;
• function passing point: (1 ; 0) .

Trigonometric functions are sine, cosine, tangent and cotangent. Let's analyze the properties of each of them and the corresponding graphs.

In general, all trigonometric functions are characterized by the property of periodicity, i.e. when the values ​​of the functions are repeated for different values ​​of the argument that differ from each other by the value of the period f (x + T) = f (x) (T is the period). Thus, the item "least positive period" is added to the list of properties of trigonometric functions. In addition, we will indicate such values ​​of the argument for which the corresponding function vanishes.

1. Sine function: y = sin(x)

The graph of this function is called a sine wave.

Definition 18

Properties of the sine function:

• domain of definition: the whole set of real numbers x ∈ - ∞ ; +∞ ;
• the function vanishes when x = π k , where k ∈ Z (Z is the set of integers);
• the function is increasing for x ∈ - π 2 + 2 π · k ; π 2 + 2 π k , k ∈ Z and decreasing for x ∈ π 2 + 2 π k ; 3 π 2 + 2 π k , k ∈ Z ;
• the sine function has local maxima at the points π 2 + 2 π · k ; 1 and local minima at points - π 2 + 2 π · k ; - 1 , k ∈ Z ;
• the sine function is concave when x ∈ - π + 2 π k; 2 π k , k ∈ Z and convex when x ∈ 2 π k ; π + 2 π k , k ∈ Z ;
• there are no asymptotes.

1. cosine function: y=cos(x)

The graph of this function is called a cosine wave.

Definition 19

Properties of the cosine function:

• domain of definition: x ∈ - ∞ ; +∞ ;
• the smallest positive period: T \u003d 2 π;
• range: y ∈ - 1 ; one ;
• this function is even, since y (- x) = y (x) ;
• the function is increasing for x ∈ - π + 2 π · k ; 2 π · k , k ∈ Z and decreasing for x ∈ 2 π · k ; π + 2 π k , k ∈ Z ;
• the cosine function has local maxima at points 2 π · k ; 1 , k ∈ Z and local minima at the points π + 2 π · k ; - 1 , k ∈ z ;
• the cosine function is concave when x ∈ π 2 + 2 π · k ; 3 π 2 + 2 π k , k ∈ Z and convex when x ∈ - π 2 + 2 π k ; π 2 + 2 π · k , k ∈ Z ;
• inflection points have coordinates π 2 + π · k ; 0 , k ∈ Z
• there are no asymptotes.

1. Tangent function: y = t g (x)

The graph of this function is called tangentoid.

Definition 20

Properties of the tangent function:

• domain of definition: x ∈ - π 2 + π · k ; π 2 + π k , where k ∈ Z (Z is the set of integers);
• The behavior of the tangent function on the boundary of the domain of definition lim x → π 2 + π · k + 0 t g (x) = - ∞ , lim x → π 2 + π · k - 0 t g (x) = + ∞ . Thus, the lines x = π 2 + π · k k ∈ Z are vertical asymptotes;
• the function vanishes when x = π k for k ∈ Z (Z is the set of integers);
• range: y ∈ - ∞ ; +∞ ;
• this function is odd because y (- x) = - y (x) ;
• the function is increasing at - π 2 + π · k ; π 2 + π k , k ∈ Z ;
• the tangent function is concave for x ∈ [ π · k ; π 2 + π k) , k ∈ Z and convex for x ∈ (- π 2 + π k ; π k ] , k ∈ Z ;
• inflection points have coordinates π k; 0 , k ∈ Z ;

1. Cotangent function: y = c t g (x)

The graph of this function is called the cotangentoid. .

Definition 21

Properties of the cotangent function:

• domain of definition: x ∈ (π k ; π + π k) , where k ∈ Z (Z is the set of integers);

Behavior of the cotangent function on the boundary of the domain of definition lim x → π · k + 0 t g (x) = + ∞ , lim x → π · k - 0 t g (x) = - ∞ . Thus, the lines x = π k k ∈ Z are vertical asymptotes;

• the smallest positive period: T \u003d π;
• the function vanishes when x = π 2 + π k for k ∈ Z (Z is the set of integers);
• range: y ∈ - ∞ ; +∞ ;
• this function is odd because y (- x) = - y (x) ;
• the function is decreasing for x ∈ π · k ; π + π k , k ∈ Z ;
• the cotangent function is concave for x ∈ (π k ; π 2 + π k ] , k ∈ Z and convex for x ∈ [ - π 2 + π k ; π k) , k ∈ Z ;
• inflection points have coordinates π 2 + π · k ; 0 , k ∈ Z ;
• there are no oblique and horizontal asymptotes.

The inverse trigonometric functions are the arcsine, arccosine, arctangent, and arccotangent. Often, due to the presence of the prefix "arc" in the name, inverse trigonometric functions are called arc functions. .

1. Arcsine function: y = a r c sin (x)

Definition 22

Properties of the arcsine function:

• this function is odd because y (- x) = - y (x) ;
• the arcsine function is concave for x ∈ 0; 1 and convexity for x ∈ - 1 ; 0;
• inflection points have coordinates (0 ; 0) , it is also the zero of the function;
• there are no asymptotes.

1. Arccosine function: y = a r c cos (x)

Definition 23

Arccosine function properties:

• domain of definition: x ∈ - 1 ; one ;
• range: y ∈ 0 ; π;
• this function is of general form (neither even nor odd);
• the function is decreasing on the entire domain of definition;
• the arccosine function is concave for x ∈ - 1 ; 0 and convexity for x ∈ 0 ; one ;
• inflection points have coordinates 0 ; π2;
• there are no asymptotes.

1. Arctangent function: y = a r c t g (x)

Definition 24

Arctangent function properties:

• domain of definition: x ∈ - ∞ ; +∞ ;
• range: y ∈ - π 2 ; π2;
• this function is odd because y (- x) = - y (x) ;
• the function is increasing over the entire domain of definition;
• the arctangent function is concave for x ∈ (- ∞ ; 0 ] and convex for x ∈ [ 0 ; + ∞) ;
• the inflection point has coordinates (0; 0), it is also the zero of the function;
• horizontal asymptotes are straight lines y = - π 2 for x → - ∞ and y = π 2 for x → + ∞ (the asymptotes in the figure are green lines).

1. Arc cotangent function: y = a r c c t g (x)

Definition 25

Arc cotangent function properties:

• domain of definition: x ∈ - ∞ ; +∞ ;
• range: y ∈ (0 ; π) ;
• this function is of a general type;
• the function is decreasing on the entire domain of definition;
• the arc cotangent function is concave for x ∈ [ 0 ; + ∞) and convexity for x ∈ (- ∞ ; 0 ] ;
• the inflection point has coordinates 0 ; π2;
• horizontal asymptotes are straight lines y = π at x → - ∞ (green line in the drawing) and y = 0 at x → + ∞.

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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 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.

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

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.

See also:

A power function is called a function of the form y=x n (read as y equals x to the power of n), where n is some given number. Particular cases of power functions are functions of the form y=x, y=x 2 , y=x 3 , y=1/x and many others. Let's talk more about each of them.

Linear function y=x 1 (y=x)

The graph is a straight line passing through the point (0;0) at an angle of 45 degrees to the positive direction of the Ox axis.

The chart is shown below.

Basic properties of a linear function:

• The function is increasing and is defined on the whole number axis.
• It has no maximum and minimum values.

Quadratic function y=x 2

The graph of a quadratic function is a parabola.

Basic properties of a quadratic function:

• 1. For x=0, y=0, and y>0 for x0
• 2. The quadratic function reaches its minimum value at its vertex. Ymin at x=0; It should also be noted that the maximum value of the function does not exist.
• 3. The function decreases on the interval (-∞; 0] and increases on the interval )