What is the graph of a power function? Power function, its properties and graph, presentation for an algebra lesson (grade 10) on the topic
Provides reference data for exponential function- basic properties, graphs and formulas. The following issues are considered: domain of definition, set of values, monotonicity, inverse function, derivative, integral, power series expansion and representation using complex numbers.
ContentProperties of the Exponential Function
The exponential function y = a x has the following properties on the set of real numbers ():
(1.1)
defined and continuous, for , for all ;
(1.2)
for a ≠ 1
has many meanings;
(1.3)
strictly increases at , strictly decreases at ,
is constant at ;
(1.4)
at ;
at ;
(1.5)
;
(1.6)
;
(1.7)
;
(1.8)
;
(1.9)
;
(1.10)
;
(1.11)
,
.
Other useful formulas.
.
Formula for converting to an exponential function with a different exponent base:
When b = e, we obtain the expression of the exponential function through the exponential:
Private values
, , , , .
y = a x for different values of the base a. The figure shows graphs of the exponential function
y (x) = ax
for four values degree bases: a = 2
, a = 8
, a = 1/2
and a = 1/8
. 1
It can be seen that for a > 0
< a < 1
the exponential function increases monotonically. The larger the base of the degree a, the stronger the growth. At
the exponential function decreases monotonically. The smaller the exponent a, the stronger the decrease.
Ascending, descending
| The exponential function for is strictly monotonic and therefore has no extrema. Its main properties are presented in the table. 1 | y = a x , a > 0 < a < 1 | |
| y = ax, | - ∞ < x < + ∞ | - ∞ < x < + ∞ |
| Domain | 0 < y < + ∞ | 0 < y < + ∞ |
| Range of values | Monotone | monotonically increases |
| monotonically decreases 0 | Zeros, y = | Zeros, y = |
| No 0 | Intercept points with the ordinate axis, x = 1 | Intercept points with the ordinate axis, x = 1 |
| + ∞ | 0 | |
| 0 | + ∞ |
y =
Inverse function
The inverse of an exponential function with base a is the logarithm to base a.
.
If , then
.
If , then
Differentiation of an exponential function To differentiate an exponential function, its base must be reduced to the number e, apply the table of derivatives and the differentiation rule.
complex function
To do this you need to use the property of logarithms
.
and the formula from the derivatives table:
.
Let an exponential function be given:
We bring it to the base e:
Let's apply the rule of differentiation of complex functions. To do this, introduce the variable
Then
.
Since is a constant, the derivative of z with respect to x is equal to
.
According to the rule of differentiation of a complex function:
.
Derivative of an exponential function
.
Derivative of nth order:
.
Deriving formulas > > >
An example of differentiating an exponential function
Find the derivative of a function
Intercept points with the ordinate axis, x = 3 5 x
Solution
Let's express the base of the exponential function through the number e.
3 = e ln 3
Let's apply the rule of differentiation of complex functions. To do this, introduce the variable
.
Enter a variable
.
Let's apply the rule of differentiation of complex functions. To do this, introduce the variable
From the table of derivatives we find:
.
Because the 5ln 3 is a constant, then the derivative of z with respect to x is equal to:
.
According to the rule of differentiation of a complex function, we have:
.
Answer
Integral
Expressions using complex numbers
Consider the function complex number z:
f (z) = a z
where z = x + iy; 2 = - 1
.
i
Let us express the complex constant a in terms of modulus r and argument φ:
Let's apply the rule of differentiation of complex functions. To do this, introduce the variable
.
a = r e i φ
φ = φ The argument φ is not uniquely defined. In general,
0 + 2 πn where n is an integer. Therefore the function f(z)
.
is also not clear. Its main significance is often considered Power function is a function of the form y = xp
, where p is a given real number.
- Properties of the power function If the indicator p = 2n
- - even natural number:
- domain of definition - all real numbers, i.e. the set R;
- set of values - non-negative numbers, i.e. y ≥ 0;
- function is even;
- Properties of the power function y = x 4 p = 2n - 1
- - odd natural number:
- domain of definition - set R;
- set of values - set R;
- function is odd;
- Properties of the power function y = x 5 p = -2n , Where n
- - natural number:
- set of values - non-negative numbers, i.e. y ≥ 0;
- set of values - positive numbers y > 0;
- Properties of the power function y = 1/x 2 p = -2n , Where n
- p = -(2n - 1)
- domain of definition - set R, except x = 0;
- set of values - set R;
- set of values - set R, except y = 0;
- Properties of the power function y = 1/x 3 p
- - positive real non-integer number:
- domain of definition - non-negative numbers x ≥ 0;
- set of values - non-negative numbers y ≥ 0;
- Properties of the power function y = 1/x 3 y = x 4/3
- - negative real non-integer number:
- - natural number:
- domain of definition - positive numbers x > 0;
y = x -1/3 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=xp y = xp
The properties and graph of a power function significantly depend on the properties of a power with a real exponent, and in particular on the values for which x And p degree makes sense x y = 1/x 3. Let us proceed to a similar consideration of various cases depending on
exponent p.
- Index p=2n-an even natural number.
properties:
- - even natural number:
- set of values - non-negative numbers, i.e. y is greater than or equal to 0;
- function y=x2n even, because x 2n=(- x) 2n
- the function is decreasing on the interval x<0 and increasing on the interval x>0.
2. Indicator p=2n-1- odd natural number
In this case, the power function y=x2n-1, where is a natural number, has the following properties:
- - odd natural number:
- domain of definition - set R;
- function y=x2n-1 odd, since (- x) 2n-1=x2n-1;
- function is odd;
3.Indicator p=-2n, Where n- natural number.
In this case, the power function y=x -2n =1/x 2n has the following properties:
- domain of definition - set R, except x=0;
- set of values - positive numbers y>0;
- function y =1/x2n even, because 1/(-x)2n=1/x 2n;
- the function is increasing on the interval x<0 и убывающей на промежутке x>0.
Basic 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 gives key material on the topic of basic elementary functions. We will introduce terms, give them definitions; Let's study each type of elementary functions in detail and analyze their properties.
The following types of basic elementary functions are distinguished:
Definition 1
- constant function (constant);
- nth root;
- power function;
- exponential function;
- logarithmic function;
- trigonometric functions;
- fraternal trigonometric functions.
A constant function is defined by the formula: y = C (C is a certain real number) and also has a name: constant. This function matches any actual value independent variable x of the same value of variable y – value C .
The graph of a constant is a straight line that is parallel to the abscissa 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 (indicated in black, red and blue colors in the drawing, respectively).
Definition 2
This elementary function is defined by the formula y = x n (n is a natural number greater than one).
Let's consider two variations of the function.
- nth root, n – even number
For clarity, we indicate a drawing that shows graphs of such functions: y = x, y = x 4 and y = x8. These features are color coded: black, red and blue respectively.
The graphs of a function of even degree have a similar appearance for other values of the exponent.
Definition 3
Properties of the nth root function, n is an even number
- domain of definition – 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 for even root exponents increases throughout the entire domain of definition;
- the function has a convexity with an upward direction throughout 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).
- nth root, n – odd number
Such a function is defined on the entire set of real numbers. For clarity, consider the graphs of the functions y = x 3 , y = x 5 and x 9 . In the drawing they are indicated by colors: black, red and Blue colour and curves respectively.
Other odd values of the root exponent of the function y = x n will give a graph of a similar type.
Definition 4
Properties of the nth root function, n is an odd number
- domain of definition – the set of all real numbers;
- this function is odd;
- range of values – the set of all real numbers;
- the function y = x n for odd root exponents 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).
is also not clear. Its main significance is often considered
Definition 5The power function is defined by the formula y = x a.
The appearance of the graphs and the properties of the function depend on the value of the exponent.
- when a power function has an integer exponent a, then the type of graph of the power function and its properties depend on whether the exponent is even or odd, as well as what sign the exponent has. Let's consider all these special cases in more detail below;
- the exponent can be fractional or irrational - depending on this, the type of graphs and 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 an odd positive number, for example, a = 1, 3, 5...
For clarity, we indicate the graphs of such power functions: y = x (graphic color black), y = x 3 (blue color of the graph), y = x 5 (red color of the graph), y = x 7 (graphic color green). When a = 1, we get linear function y = x.
Definition 6
Properties of a power function when the exponent is odd positive
- the function is increasing for x ∈ (- ∞ ; + ∞) ;
- the function has convexity for x ∈ (- ∞ ; 0 ] and concavity for x ∈ [ 0 ; + ∞) (excluding the linear function);
- the inflection point has coordinates (0 ; 0) (excluding linear function);
- there are no asymptotes;
- points of passage of the function: (- 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 = x 2 (graphic color black), y = x 4 (blue color of the graph), y = x 8 (red color of the graph). When a = 2, we get quadratic function, the graph of which 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 has concavity for x ∈ (- ∞ ; + ∞) ;
- there are no inflection points;
- there are no asymptotes;
- points of passage of the function: (- 1 ; 1) , (0 ; 0) , (1 ; 1) .
The figure below shows examples of power function graphs y = x a when a is an odd negative number: y = x - 9 (graphic color black); y = x - 5 (blue color of the graph); y = x - 3 (red color of the graph); y = x - 1 (graphic color green). When a = - 1, we obtain inverse proportionality, the graph of which is a hyperbola.
Definition 8
Properties of a power function when the exponent is odd negative:
When x = 0, we obtain a discontinuity of the second kind, since lim x → 0 - 0 x a = - ∞, lim x → 0 + 0 x a = + ∞ for a = - 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 has convexity for x ∈ (- ∞ ; 0) and concavity 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, . . . .
- points of passage of the function: (- 1 ; - 1) , (1 ; 1) .
The figure below shows examples of graphs of the power function y = x a when a is an even negative number: y = x - 8 (graphic color black); y = x - 4 (blue color of the graph); y = x - 2 (red color of the graph).
Definition 9
Properties of a power function when the exponent is even negative:
- domain of definition: x ∈ (- ∞ ; 0) ∪ (0 ; + ∞) ;
When x = 0, we obtain a discontinuity of the second kind, since lim x → 0 - 0 x a = + ∞, lim x → 0 + 0 x a = + ∞ for a = - 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 has concavity at x ∈ (- ∞ ; 0) ∪ (0 ; + ∞) ;
- there are no inflection points;
- horizontal asymptote – 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 , . . . .
- points of passage of the function: (- 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. On this moment authors of many educational publications in algebra and the principles of analysis DO NOT DETERMINE power functions, where the exponent is a fraction with an odd denominator for negative values of the argument. Further we will adhere to exactly this position: we will take the set [ 0 ; + ∞) . Recommendation for students: find out the teacher’s view on this point in order to avoid disagreements.
So, let's 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 the power functions with graphs y = x a when a = 11 12 (graphic color black); a = 5 7 (red color of the graph); a = 1 3 (blue color of the graph); a = 2 5 (green color of the graph).
Other values of the exponent a (provided 0< a < 1) дадут аналогичный вид графика.
Definition 10
Properties of the power function at 0< a < 1:
- range: y ∈ [ 0 ; + ∞) ;
- the function is increasing for x ∈ [ 0 ; + ∞) ;
- the function is convex 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.
Let us illustrate with graphs the power function y = x a under given conditions using the following functions as an example: y = x 5 4 , y = x 4 3 , y = x 7 3 , y = x 3 π (black, red, blue, green color of the graphs, respectively).
Other values of the exponent a, provided a > 1, will give a similar graph.
Definition 11
Properties of the power function 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 has concavity for x ∈ (0 ; + ∞) (when 1< a < 2) и выпуклость при x ∈ [ 0 ; + ∞) (когда a > 2);
- there are no inflection points;
- there are no asymptotes;
- points of passage of the function: (0 ; 0) , (1 ; 1) .
Please note! 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 is in in this case– interval - ∞; 0 ∪ (0 ; + ∞) with the caveat that the exponent a is an irreducible fraction. Currently the authors educational materials in algebra and principles of analysis DO NOT DETERMINE power functions with an exponent in the form of a fraction with an odd denominator for negative values of the argument. Further, we adhere to exactly this view: we take the set (0 ; + ∞) as the domain of definition of power functions with fractional negative exponents. Recommendation for students: Clarify your teacher's vision at this point to avoid disagreements.
Let's continue the topic and analyze the power function y = x a provided: - 1< a < 0 .
Let us present 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 color of the lines, respectively).
Definition 12
Properties of the power function 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
Properties of the power function 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 has concavity for x ∈ 0; + ∞ ;
- there are no inflection points;
- horizontal asymptote – straight line y = 0;
- point of passage of the function: (1; 1) .
When a = 0 and x ≠ 0, we obtain the function y = x 0 = 1, which defines the line from which the point (0; 1) is excluded (it was agreed that the expression 0 0 will not be given any meaning).
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 look at the situation when the base of the exponential function has a value from zero to one (0< a < 1) . A good 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 appearance for other values of the base under the condition 0< a < 1 .
Definition 14
Properties of the 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;
- horizontal asymptote – straight line y = 0 with variable x tending to + ∞;
Now consider the case when the base of the exponential function is greater than one (a > 1).
Let us illustrate this special case with a 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, larger units, will give a similar appearance to the graph of the exponential function.
Definition 15
Properties of the exponential function when the base is greater than one:
- domain of definition – 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 as x ∈ - ∞; + ∞ ;
- the function has a concavity at x ∈ - ∞; + ∞ ;
- there are no inflection points;
- horizontal asymptote – straight line y = 0 with variable x tending to - ∞;
- point of passage of the function: (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 graph of a logarithmic function has different kind, based on the value of base a.
Let us first consider 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 larger units, will give a similar type of 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 function values tend to +∞;
- range: y ∈ - ∞ ; + ∞ ;
- this function is a function of general form (it is neither odd nor even);
- logarithmic
- the function has concavity for x ∈ 0; + ∞ ;
- there are no inflection points;
- there are no asymptotes;
Now let’s look at the special case when the base of the logarithmic function is greater than one: a > 1 . The drawing below shows 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 type of 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 function values tend to - ∞ ;
- range: y ∈ - ∞ ; + ∞ (the entire 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 is convex for x ∈ 0; + ∞ ;
- there are no inflection points;
- there are no asymptotes;
- point of passage of the function: (1; 0) .
The trigonometric functions are sine, cosine, tangent and cotangent. Let's look at the properties of each of them and the corresponding graphics.
In general, all trigonometric functions are characterized by the property of periodicity, i.e. when function values are repeated at different meanings arguments differing from each other by the period f (x + T) = f (x) (T – period). Thus, the item “smallest positive period” is added to the list of properties of trigonometric functions. In addition, we will indicate the values of the argument at which the corresponding function becomes zero.
- 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 entire 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 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.
- 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 ∈ - ∞ ; + ∞ ;
- smallest positive period: T = 2 π;
- range of values: y ∈ - 1 ; 1 ;
- 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 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.
- Tangent function: y = t g (x)
The graph of this function is called tangent.
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);
- 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 straight 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, since y (- x) = - y (x) ;
- the function is increasing as - 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 ;
- Cotangent function: y = c t g (x)
The graph of this function is called a 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 straight lines x = π · k k ∈ Z are vertical asymptotes;
- smallest positive period: T = 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, since 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 ;
- inclined and horizontal asymptotes none.
The inverse trigonometric functions are arcsine, arccosine, arctangent and arccotangent. Often, due to the presence of the prefix “arc” in the name, inverse trigonometric functions are called arc functions .
- Arc sine function: y = a r c sin (x)
Definition 22
Properties of the arcsine function:
- this function is odd, since y (- x) = - y (x) ;
- the arcsine function has a concavity for x ∈ 0; 1 and convexity for x ∈ - 1 ; 0 ;
- inflection points have coordinates (0; 0), which is also the zero of the function;
- there are no asymptotes.
- Arc cosine function: y = a r c cos (x)
Definition 23
Properties of the arc cosine function:
- domain of definition: x ∈ - 1 ; 1 ;
- range: y ∈ 0 ; π;
- this function is of a general form (neither even nor odd);
- the function is decreasing over the entire domain of definition;
- the arc cosine function has a concavity at x ∈ - 1; 0 and convexity for x ∈ 0; 1 ;
- inflection points have coordinates 0; π 2;
- there are no asymptotes.
- Arctangent function: y = a r c t g (x)
Definition 24
Properties of the arctangent function:
- domain of definition: x ∈ - ∞ ; + ∞ ;
- range of values: y ∈ - π 2 ; π 2;
- this function is odd, since y (- x) = - y (x) ;
- the function is increasing over the entire domain of definition;
- the arctangent function has concavity for x ∈ (- ∞ ; 0 ] and convexity for x ∈ [ 0 ; + ∞);
- the inflection point has coordinates (0; 0), which is also the zero of the function;
- horizontal asymptotes are straight lines y = - π 2 as x → - ∞ and y = π 2 as x → + ∞ (in the figure, the asymptotes are green lines).
- Arc tangent function: y = a r c c t g (x)
Definition 25
Properties of the arccotangent function:
- domain of definition: x ∈ - ∞ ; + ∞ ;
- range: y ∈ (0; π) ;
- this function is of a general form;
- the function is decreasing over the entire domain of definition;
- the arc cotangent function has a concavity for x ∈ [ 0 ; + ∞) and convexity for x ∈ (- ∞ ; 0 ] ;
- the inflection point has coordinates 0; π 2;
- horizontal asymptotes are straight lines y = π at x → - ∞ (green line in the drawing) and y = 0 at x → + ∞.
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Are you familiar with the functions 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 y = 1/x 3, where p is a given real number. The properties and graph of a power function significantly depend on the properties of a power with a real exponent, and in particular on the values for which x And y = 1/x 3 degree makes sense x y = 1/x 3. Let us proceed to a similar consideration of various cases depending on the exponent p.
Index p=2n-an even natural number.
In this case, the power function y=x 2n, Where , Where- a natural number, has the following
properties:
domain of definition - 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.
Graph of a function 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:
- odd natural number:
domain of definition - set R;
function y=x 2n-1 odd, since (- x) 2n-1 =x 2n-1 ;
the function is increasing on the entire real axis.
Graph of a function y=x2n-1 has the same form as, for example, the graph of a 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 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 , Where- natural number. In this case, the power function y=x -(2n-1) has the following properties:
domain of definition - set R, except x=0;
set of values - set R, except y=0;
function y=x -(2n-1) odd, since (- x) -(2n-1) =-x -(2n-1) ;
the function is decreasing on intervals x<0 And x>0.
Graph of a function y=x -(2n-1) has the same form as, for example, the graph of a function y=1/x 3 .
Inverse trigonometric functions, their properties and graphs.
Inverse trigonometric functions, their properties and graphs.Inverse trigonometric functions (circular functions, arc functions) - mathematical functions that are the inverse of trigonometric functions.
arcsin function
Graph of a function 
.
arcsine numbers m this angle value is called x, for which
The function is continuous and bounded along its entire number line. Function 
is strictly increasing.
[Edit]Properties of the arcsin function
[Edit]Getting the arcsin function
Given the function Throughout its entire domain of definition she happens to be piecewise monotonic, and, therefore, the inverse correspondence 
is not a function. Therefore, we will consider the segment on which it strictly increases and takes on all values range of values- . Since for a function on an interval each value of the argument corresponds to a single value of the function, then on this interval there is 
inverse function