Table of basic properties of elementary functions. Functions and graphs

1) The domain of the function and the domain of the function.

Function scope is the set of all valid valid argument values x(variable x) for which the function y = f (x) defined. The range of values of a function is the set of all real values y that the function accepts.

In elementary mathematics, functions are studied only on the set of real numbers.

2) Function zeros.

Function zero is an argument value at which the function value is equal to zero.

3) Intervals of constancy of function.

The intervals of constant sign of a function are such sets of argument values, on which the values of the function are only positive or only negative.

4) Monotonicity of function.

An increasing function (in a certain interval) is a function for which a larger value of the argument from this interval corresponds to a larger value of the function.

Decreasing function (in a certain interval) - a function in which the larger value of the argument from this interval corresponds to the smaller value of the function.

5) Parity (odd) function.

An even function is a function whose domain of definition is symmetric about the origin and for any X from the domain, the equality f (-x) = f (x)... The graph of an even function is symmetric about the ordinate axis.

An odd function is a function whose domain of definition is symmetric about the origin and for any X the domain of definition satisfies the equality f (-x) = - f (x). The graph of an odd function is symmetric about the origin.

6) Limited and unlimited functions.

A function is called bounded if there exists a positive number M such that | f (x) | ≤ M for all values of x. If there is no such number, then the function is unlimited.

7) Periodicity of function.

A function f (x) is periodic if there is a nonzero number T such that for any x from the domain of the function the following holds: f (x + T) = f (x). This smallest number is called the period of the function. All trigonometric functions are periodic. (Trigonometric formulas).

19. Basic elementary functions, their properties and graphics. Application of functions in economics.

Basic elementary functions. Their properties and graphs

1. Linear function.

Linear function called a function of the form, where x is a variable, a and b are real numbers.

Number a called the slope of a straight line, it is equal to the tangent of the angle of inclination of this straight line to the positive direction of the abscissa axis. The graph of a linear function is a straight line. It is defined by two points.

Linear function properties

1. Domain of definition - the set of all real numbers: D (y) = R

2. The set of values is the set of all real numbers: E (y) = R

3. The function takes on a zero value for or.

4. The function increases (decreases) over the entire domain of definition.

5. The linear function is continuous on the whole domain of definition, differentiable and.

2. Quadratic function.

A function of the form, where x is a variable, the coefficients a, b, c are real numbers, is called quadratic.

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 will list all the basic elementary functions, give their graphs and give them without derivation and proof. properties of basic elementary functions according to the scheme:

the behavior of the function on the boundaries of the domain of definition, vertical asymptotes (if necessary, see the article on the classification of breakpoints of a function);
even and odd;
intervals of convexity (convexity upward) and concavity (convexity downward), inflection points (if necessary, see the article convexity of a function, direction of convexity, inflection points, convexity and inflection conditions);
oblique and horizontal asymptotes;
special 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), n-th root, power function, exponential, logarithmic function, trigonometric and inverse trigonometric functions.

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Permanent function.

A constant function is defined 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 of C. A constant function is also called a constant.

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

Constant function properties.

Domain of definition: the whole set of real numbers.
The constant function is even.
Range of values: a set consisting of the singular number C.
The 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 a constant.
There are no asymptotes.
The function passes through the point (0, C) of the coordinate plane.

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.

Nth root, n is an even number.

Let's start with the function n-th root for even values of the exponent of the root n.

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

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

Properties of the nth root function for even n.

Nth root, n is an odd number.

The function n-th root with an odd exponent of the root n is defined on the entire set of real numbers. For example, we will give graphs of functions and, they correspond to the black, red and blue curves.

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

Properties of the nth root function for odd n.

Power function.

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

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

Let's start with a power function with integer exponent a. In this case, the form of graphs of power functions and the properties of the functions depend on the evenness or oddness of the 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 exponents, 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 form of graphs of such power functions) depend on the value of the exponent a. They will be considered, firstly, for a from zero to one, secondly, when a is large units, thirdly, when a is from minus one to zero, and fourthly, when a is less than minus one.

To conclude this subsection, for completeness, we describe a power function with zero exponent.

Power function with an 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-law 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, with a = 2,4,6,….

As an example, we will give 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.

A power function with an odd negative exponent.

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

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

Properties of a power function with an odd negative exponent.

An even negative exponent.

Let us pass to the power function for 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.

Power function with a rational or irrational exponent, the value of which 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 definition 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 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. We will adhere to just such a view, that is, we will consider the domains of definition of power functions with fractional positive exponents of the set. We encourage students to get to know your teacher's perspective on this delicate point in order to avoid controversy.

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

Here are the graphs of power functions for a = 11/12 (black line), a = 5/7 (red line), (blue line), and = 2/5 (green line).

Power function with non-integral rational or irrational exponent greater than one.

Consider a power function with noninteger 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 look similar.

Power function properties for.

Power function with real exponent 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 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. We will adhere to just such a view, that is, we will consider the domains of definition of power functions with fractional negative fractional exponents, respectively. We encourage students to get to know your teacher's perspective on this delicate point in order to avoid controversy.

We pass to the power function, kg.

In order to have a good idea of the form 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 less than minus one.

Let us give examples of graphs of power functions for , they are depicted with 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 (we agreed not to attach any meaning to the expression 0 0).

Exponential function.

One of the basic elementary functions is the exponential function.

The exponential function graph, where and takes on 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 will give graphs of the exponential function with a = 1/2 - the blue line, a = 5/6 - the red line. The plots of the exponential function for other values of the base from the interval have a similar form.

Properties of exponential function with 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 - blue line and - red line. For other values of the base, greater than one, the graphs of the exponential function will have a similar appearance.

Properties of the 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.

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

Yandex.RTB R-A-339285-1

The article below provides key material on the topic of basic elementary functions. We will introduce terms, define them; we will study in detail each type of elementary functions, we will 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 valid value of the independent variable x corresponds to the same value of the variable y - the value of C.

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

Definition 2

This elementary function is determined by the formula y = x n (n is a natural number greater than one).

Consider two variations of the function.

Nth root, 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.

The graphs of the function of even degree have a similar look at other values of the indicator.

Definition 3

Properties of the function nth root, 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;
this function is a general function (it is neither even nor odd);
range of values: [0, + ∞);
this function y = x n for even root exponents increases over the entire domain of definition;
the function has a convexity with an upward direction on 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 is an odd number

This 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 the 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 type.

Definition 4

Properties of the function nth root, 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 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 determined by the formula y = x a.

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

when the 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, as well as 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 ;
the power function can have a zero exponent; we will also analyze this case in more detail below.

Let us 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 (black color of the chart), y = x 3 (blue color of the graph), y = x 5 (red color of the graph), y = x 7 (green color of the graph). When a = 1, we get a 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 is convex for x ∈ (- ∞; 0] and concavity for x ∈ [0; + ∞) (excluding the linear function);
the inflection point has coordinates (0; 0) (excluding the linear function);
there are no asymptotes;
the points of passage of the function: (- 1; - 1), (0; 0), (1; 1).

Let us 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 (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 ∈ (- ∞; + ∞);
no inflection points;
there are no asymptotes;
the points of passage of the function: (- 1; 1), (0; 0), (1; 1).

The figure below shows examples of graphs of a power function y = x a when a is an odd negative number: y = x - 9 (black color of the graph); y = x - 5 (blue color of the graph); y = x - 3 (red color of the graph); y = x - 1 (green color of the graph). When a = - 1, we get 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 get 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 the vertical asymptote;

range of values: 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 concavity for x ∈ (0; + ∞);
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,. ... ... ...

the points of passage of the function: (- 1; - 1), (1; 1).

The figure below shows examples of graphs of a power function y = x a, when a is an even negative number: y = x - 8 (black color of the graph); 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: x ∈ (- ∞; 0) ∪ (0; + ∞);

When x = 0, we get 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 the 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; + ∞);
no inflection points;
the horizontal asymptote is the 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,. ... ... ...

the 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; + ∞, while stipulating that the exponent a is an irreducible fraction. At the moment, the authors of many educational publications on 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 just this position: we will take as the domain of definition of power functions with fractional positive exponents the set [0; + ∞). Tip for students: Find out the teacher's perspective at this point in order to avoid controversy.

So, let's analyze 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 (black color of the graph); 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 that 0< a < 1) дадут аналогичный вид графика.

Definition 10

Power function properties at 0< a < 1:

range of values: y ∈ [0; + ∞);
the function is increasing for x ∈ [0; + ∞);
the function is convex for x ∈ (0; + ∞);
no inflection points;
there are no asymptotes;

Let us 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 the given conditions using 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 graphs, respectively).

Other values of the exponent a, provided a> 1, will give a similar view of the graph.

Definition 11

Power function properties for a> 1:

domain of definition: x ∈ [0; + ∞);
range of values: y ∈ [0; + ∞);
this function is a general function (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);
no inflection points;
there are no asymptotes;
the points of passage of the function: (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 is the interval - ∞; 0 ∪ (0; + ∞) with the proviso that the exponent a is an irreducible fraction. At the moment, the authors of educational materials on algebra and the 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 just such a view: we will take the set (0; + ∞) as the domain of definition of power functions with fractional negative exponents. Tip for students: Clarify your teacher's vision at this point to avoid controversy.

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

Here is a drawing of the 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 of values: y ∈ 0; + ∞;
this function is a general function (it is neither odd nor even);
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 of the 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 of values: y ∈ (0; + ∞);
this function is a general function (it is neither odd nor even);
the function is decreasing for x ∈ 0; + ∞;
the function is concave for x ∈ 0; + ∞;
no inflection points;
horizontal asymptote - straight line y = 0;
function passage point: (1; 1).

When a = 0 and x ≠ 0, we get the function y = x 0 = 1, which defines the line from which the point (0; 1) is excluded (we 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 us 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 plots of the exponential function will have a similar form for other values of the base, provided that 0< a < 1 .

Definition 14

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

range of values: y ∈ (0; + ∞);
this function is a general function (it is neither odd nor even);
the exponential function, for which the base is less than one, is decreasing over the entire domain of definition;
no inflection points;
horizontal asymptote - 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 particular 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 view of the exponential function graph.

Definition 15

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

domain of definition - the whole set of real numbers;
range of values: y ∈ (0; + ∞);
this function is a general function (it is neither odd nor even);
an exponential function with a base greater than one is increasing for x ∈ - ∞; + ∞;
the function is concave for x ∈ - ∞; + ∞;
no inflection points;
horizontal asymptote - line y = 0 with variable x tending to - ∞;
function passage 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 graph of the logarithmic function has a different form, based on the value of the 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 large units, will give a similar view of the chart.

Definition 16

Properties of the 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 of values: y ∈ - ∞; + ∞;
this function is a general function (it is neither odd nor even);
logarithmic
the function is concave for x ∈ 0; + ∞;
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, the graphs of the logarithmic functions y = log 3 2 x and y = ln x (blue and red colors of the graphs, respectively).

Other base values greater than one will give a similar chart appearance.

Definition 17

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

domain of definition: x ∈ 0; + ∞. When x tends to zero from the right, the values of the function tend to - ∞;
range of values: y ∈ - ∞; + ∞ (all set of real numbers);
this function is a general function (it is neither odd nor even);
the logarithmic function is increasing for x ∈ 0; + ∞;
the function is convex for x ∈ 0; + ∞;
no inflection points;
there are no asymptotes;
function passage 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 "smallest 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.

Sine function: y = sin (x)

The graph of this function is called a sine wave.

Definition 18

Sine function properties:

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 function is sine-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

Cosine function properties:

domain of definition: x ∈ - ∞; + ∞;
the smallest positive period: T = 2 π;
range of values: 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 the 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.

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);
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 of values: 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: 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 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;

the smallest positive period: T = π;
the function vanishes when x = π 2 + π · k for k ∈ Z (Z is the set of integers);
range of values: 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;
oblique and horizontal asymptotes are absent.

Inverse trigonometric functions are inverse sine, inverse cosine, arctangent, and inverse cotangent. Often, due to the presence of the prefix "arc" in the name, inverse trigonometric functions are called arc functions .

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

Definition 22

Arcsine function properties:

this function is odd since y (- x) = - y (x);
the arcsine function has concavity for x ∈ 0; 1 and convexity for x ∈ - 1; 0;
inflection points have coordinates (0; 0), which is the zero of the function;
there are no asymptotes.

Arc cosine function: y = a r c cos (x)

Definition 23

Properties of the inverse cosine function:

domain of definition: x ∈ - 1; one ;
range of values: y ∈ 0; π;
this function is of a general type (neither even nor odd);
the function is decreasing over the entire domain of definition;
the inverse cosine function has concavity for x ∈ - 1; 0 and convexity for x ∈ 0; one ;
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), it 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, asymptotes are green lines).

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

Definition 25

Properties of the inverse cotangent function:

domain of definition: x ∈ - ∞; + ∞;
range of values: y ∈ (0; π);
this function is of a general type;
the function is decreasing over the entire domain of definition;
the arc cotangent function has concavity for x ∈ [0; + ∞) and convexity for x ∈ (- ∞; 0];
the inflection point has coordinates 0; π 2;
horizontal asymptotes are straight lines y = π as x → - ∞ (in the drawing - a green line) and y = 0 as x → + ∞.

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