What is a function table. Construction of graphs of functions. Power function with even positive exponent

1. Linear fractional function and its graph

A function of the form y = P(x) / Q(x), where P(x) and Q(x) are polynomials, is called a fractional rational function.

You are probably already familiar with the concept of rational numbers. Similarly rational functions are functions that can be represented as a quotient of two polynomials.

If a fractional rational function is a quotient of two linear functions - polynomials of the first degree, i.e. view function

y = (ax + b) / (cx + d), then it is called fractional linear.

Note that in the function y = (ax + b) / (cx + d), c ≠ 0 (otherwise the function becomes linear y = ax/d + b/d) and that a/c ≠ b/d (otherwise the function is a constant ). The linear-fractional function is defined for all real numbers, except for x = -d/c. Graphs of linear-fractional functions do not differ in form from the graph you know y = 1/x. The curve that is the graph of the function y = 1/x is called hyperbole. With an unlimited increase in x in absolute value, the function y = 1/x decreases indefinitely in absolute value and both branches of the graph approach the abscissa axis: the right one approaches from above, and the left one approaches from below. The lines approached by the branches of a hyperbola are called its asymptotes.

Example 1

y = (2x + 1) / (x - 3).

Solution.

Let's select the integer part: (2x + 1) / (x - 3) = 2 + 7 / (x - 3).

Now it is easy to see that the graph of this function is obtained from the graph of the function y = 1/x by the following transformations: shift by 3 unit segments to the right, stretch along the Oy axis by 7 times and shift by 2 unit segments up.

Any fraction y = (ax + b) / (cx + d) can be written in the same way, highlighting the “whole part”. Consequently, the graphs of all linear-fractional functions are hyperbolas shifted along the coordinate axes in various ways and stretched along the Oy axis.

To plot a graph of some arbitrary linear-fractional function, it is not at all necessary to transform the fraction that defines this function. Since we know that the graph is a hyperbola, it will be enough to find the lines to which its branches approach - the hyperbola asymptotes x = -d/c and y = a/c.

Example 2

Find the asymptotes of the graph of the function y = (3x + 5)/(2x + 2).

Solution.

The function is not defined, for x = -1. Hence, the line x = -1 serves as a vertical asymptote. To find the horizontal asymptote, let's find out what the values ​​of the function y(x) approach when the argument x increases in absolute value.

To do this, we divide the numerator and denominator of the fraction by x:

y = (3 + 5/x) / (2 + 2/x).

As x → ∞ the fraction tends to 3/2. Hence, the horizontal asymptote is the straight line y = 3/2.

Example 3

Plot the function y = (2x + 1)/(x + 1).

Solution.

We select the “whole part” of the fraction:

(2x + 1) / (x + 1) = (2x + 2 - 1) / (x + 1) = 2(x + 1) / (x + 1) - 1/(x + 1) =

2 – 1/(x + 1).

Now it is easy to see that the graph of this function is obtained from the graph of the function y = 1/x by the following transformations: a shift of 1 unit to the left, a symmetric display with respect to Ox, and a shift of 2 unit intervals up along the Oy axis.

Domain of definition D(y) = (-∞; -1)ᴗ(-1; +∞).

Range of values ​​E(y) = (-∞; 2)ᴗ(2; +∞).

Intersection points with axes: c Oy: (0; 1); c Ox: (-1/2; 0). The function increases on each of the intervals of the domain of definition.

Answer: figure 1.

2. Fractional-rational function

Consider a fractional rational function of the form y = P(x) / Q(x), where P(x) and Q(x) are polynomials of degree higher than the first.

Examples of such rational functions:

y \u003d (x 3 - 5x + 6) / (x 7 - 6) or y \u003d (x - 2) 2 (x + 1) / (x 2 + 3).

If the function y = P(x) / Q(x) is a quotient of two polynomials of degree higher than the first, then its graph will, as a rule, be more complicated, and it can sometimes be difficult to build it exactly, with all the details. However, it is often enough to apply techniques similar to those with which we have already met above.

Let the fraction be proper (n< m). Известно, что любую несократимую рациональную дробь можно представить, и притом единственным образом, в виде суммы конечного числа элементарных дробей, вид которых определяется разложением знаменателя дроби Q(x) в произведение действительных сомножителей:

P(x) / Q(x) \u003d A 1 / (x - K 1) m1 + A 2 / (x - K 1) m1-1 + ... + A m1 / (x - K 1) + ... +

L 1 /(x – K s) ms + L 2 /(x – K s) ms-1 + … + L ms /(x – K s) + …+

+ (B 1 x + C 1) / (x 2 +p 1 x + q 1) m1 + … + (B m1 x + C m1) / (x 2 +p 1 x + q 1) + …+

+ (M 1 x + N 1) / (x 2 + p t x + q t) m1 + ... + (M m1 x + N m1) / (x 2 + p t x + q t).

Obviously, the graph of a fractional rational function can be obtained as the sum of graphs of elementary fractions.

Plotting fractional rational functions

Consider several ways to plot a fractional-rational function.

Example 4

Plot the function y = 1/x 2 .

Solution.

We use the graph of the function y \u003d x 2 to plot the graph y \u003d 1 / x 2 and use the method of "dividing" the graphs.

Domain D(y) = (-∞; 0)ᴗ(0; +∞).

Range of values ​​E(y) = (0; +∞).

There are no points of intersection with the axes. The function is even. Increases for all x from the interval (-∞; 0), decreases for x from 0 to +∞.

Answer: figure 2.

Example 5

Plot the function y = (x 2 - 4x + 3) / (9 - 3x).

Solution.

Domain D(y) = (-∞; 3)ᴗ(3; +∞).

y \u003d (x 2 - 4x + 3) / (9 - 3x) \u003d (x - 3) (x - 1) / (-3 (x - 3)) \u003d - (x - 1) / 3 \u003d -x / 3 + 1/3.

Here we used the technique of factorization, reduction and reduction to a linear function.

Answer: figure 3.

Example 6

Plot the function y \u003d (x 2 - 1) / (x 2 + 1).

Solution.

The domain of definition is D(y) = R. Since the function is even, the graph is symmetrical about the y-axis. Before plotting, we again transform the expression by highlighting the integer part:

y \u003d (x 2 - 1) / (x 2 + 1) \u003d 1 - 2 / (x 2 + 1).

Note that the selection of the integer part in the formula of a fractional-rational function is one of the main ones when plotting graphs.

If x → ±∞, then y → 1, i.e., the line y = 1 is a horizontal asymptote.

Answer: figure 4.

Example 7

Consider the function y = x/(x 2 + 1) and try to find exactly its largest value, i.e. the highest point on the right half of the graph. To accurately build this graph, today's knowledge is not enough. It is obvious that our curve cannot "climb" very high, since the denominator quickly begins to “overtake” the numerator. Let's see if the value of the function can be equal to 1. To do this, you need to solve the equation x 2 + 1 \u003d x, x 2 - x + 1 \u003d 0. This equation has no real roots. So our assumption is wrong. To find the largest value of the function, you need to find out for which largest A the equation A \u003d x / (x 2 + 1) will have a solution. Let's replace the original equation with a quadratic one: Ax 2 - x + A \u003d 0. This equation has a solution when 1 - 4A 2 ≥ 0. From here we find the largest value A \u003d 1/2.

Answer: Figure 5, max y(x) = ½.

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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, 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 is a natural number greater 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;
• this function is a function of general form (it 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 an 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 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 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 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 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 the moment, the authors of many educational publications on algebra and the beginnings of analysis DO NOT DEFINE power functions, 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 \u003d x a under given conditions using the following functions as an example: y \u003d x 5 4, y \u003d x 4 3, y \u003d x 7 3, y \u003d x 3 π (black, red, blue, green 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 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 beginnings of analysis DO NOT DEFINE power functions with an exponent in the form of a fraction with an odd denominator for negative values ​​of the argument. 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 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: 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 graph of the logarithmic function has a different form, 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 ; 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 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 ; 1 ;
• 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 ; 1 ;
• 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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Knowledge basic elementary functions, their properties and graphs no less important than knowing the multiplication table. They are like a foundation, everything is based on them, everything is built from them, and everything comes down to them.

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

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

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

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

Page navigation.

Permanent function.

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

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

Properties of a constant function.

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

The root of the nth degree.

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

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

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

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

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

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

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

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

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

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

Power function.

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

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

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

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

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

Power function with odd positive exponent.

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

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

Properties of a power function with an odd positive exponent.

Power function with even positive exponent.

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

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

Properties of a power function with an even positive exponent.

Power function with an odd negative exponent.

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

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

Properties of a power function with an odd negative exponent.

Power function with even negative exponent.

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

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

Properties of a power function with an even negative exponent.

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

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

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

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

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

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

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

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

Power function properties for .

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

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

We pass to the power function , where .

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

Properties of a power function with exponent a , .

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

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

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

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

Exponential function.

One of the basic elementary functions is the exponential function.

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

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

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

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

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

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

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

Logarithmic function.

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

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

Let's start with the case when .

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

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

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

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

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

Trigonometric functions, their properties and graphs.

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

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

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

The sine function y = sin(x) .

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

Properties of the sine function y = sinx .

Cosine function y = cos(x) .

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

Cosine function properties y = cosx .

Tangent function y = tg(x) .

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

Function properties tangent y = tgx .

Cotangent function y = ctg(x) .

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

Cotangent function properties y = ctgx .

Inverse trigonometric functions, their properties and graphs.

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

Arcsine function y = arcsin(x) .

Let's plot the arcsine function:

Function properties arccotangent y = arcctg(x) .

Bibliography.

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

Functions and their graphs are one of the most fascinating topics in school mathematics. It's just a pity that she passes... past the lessons and past the students. There is never enough time for her in high school. And those functions that take place in the 7th grade - a linear function and a parabola - are too simple and uncomplicated to show all the variety of interesting tasks.

The ability to build graphs of functions is necessary for solving problems with parameters on the exam in mathematics. This is one of the first topics of the course of mathematical analysis at the university. This is such an important topic that we, at the Unified State Exam-Studio, conduct special intensive courses on it for high school students and teachers, in Moscow and online. And often the participants say: “It is a pity that we did not know this before.”

But that's not all. It is with the concept of a function that real, “adult” mathematics begins. After all, addition and subtraction, multiplication and division, fractions and proportions - this is still arithmetic. Expression transformations are algebra. And mathematics is a science not only about numbers, but also about the relationships of quantities. The language of functions and graphs is understandable to a physicist, a biologist, and an economist. And as Galileo Galilei said, "The book of nature is written in the language of mathematics".

More precisely, Galileo Galilei said this: “Mathematics is the alphabet by which the Lord drew the Universe.”

Topics to review:

1. Graph the function

A familiar challenge! Such met in the variants of the OGE in mathematics. There they were considered difficult. But there is nothing complicated here.

Let's simplify the function formula:

Function graph - straight line with a punched out point

2. Graph the function

We select the integer part in the function formula:

The graph of the function is a hyperbola shifted 3 to the right in x and 2 up in y and stretched 10 times compared to the function graph

The selection of the integer part is a useful technique used in solving inequalities, plotting graphs and estimating integers in problems on numbers and their properties. You will also meet him in the first year, when you have to take integrals.

3. Graph the function

It is obtained from the graph of the function by stretching 2 times, flipping vertically and shifting 1 up vertically

4. Graph the function

The main thing is the correct sequence of actions. Let's write the function formula in a more convenient form:

We act in order:

1) Shift the graph of the function y=sinx to the left;

2) squeeze 2 times horizontally,

3) stretch 3 times vertically,

4) move up by 1

Now we will build several graphs of fractional rational functions. To better understand how we do this, read the article “Function Behavior at Infinity. Asymptotes".

5. Graph the function

Function scope:

Function zeros: and

The straight line x = 0 (y-axis) is the vertical asymptote of the function. Asymptote- a straight line, to which the graph of a function approaches infinitely close, but does not intersect it and does not merge with it (see the topic "Behavior of a function at infinity. Asymptotes")

Are there other asymptotes for our function? To find out, let's see how the function behaves as x goes to infinity.

Let's open the brackets in the function formula:

If x goes to infinity, then it goes to zero. The straight line is an oblique asymptote to the graph of the function.

6. Graph the function

This is a fractional rational function.

Function scope

Function zeros: points - 3, 2, 6.

The intervals of sign constancy of the function will be determined using the method of intervals.

Vertical asymptotes:

If x tends to infinity, then y tends to 1. Hence, is a horizontal asymptote.

Here is a sketch of the graph:

Another interesting technique is the addition of graphs.

7. Graph the function

If x tends to infinity, then the graph of the function will approach infinitely close to the oblique asymptote

If x tends to zero, then the function behaves like This is what we see on the graph:

So we have built a graph of the sum of functions. Now the work schedule!

8. Graph the function

The domain of this function is positive numbers, since only positive x is defined

The function values ​​are zero at (when the logarithm is zero), as well as at points where, that is, at

When , the value (cos x) is equal to one. The value of the function at these points will be equal to

9. Graph the function

The function is defined for It is even, since it is the product of two odd functions and The graph is symmetrical about the y-axis.

The zeros of the function are at points where, that is, at

If x goes to infinity, goes to zero. But what happens if x tends to zero? After all, both x and sin x will become smaller and smaller. How will the private behave?

It turns out that if x tends to zero, then it tends to one. In mathematics, this statement is called the "First Remarkable Limit."

But what about the derivative? Yes, we finally got there. The derivative helps to plot functions more accurately. Find maximum and minimum points, as well as function values ​​at these points.

10. Graph the function

The scope of the function is all real numbers, because

The function is odd. Its graph is symmetrical with respect to the origin.

At x=0 the value of the function is equal to zero. For the values ​​of the function are positive, for are negative.

If x goes to infinity, then it goes to zero.

Let's find the derivative of the function
According to the formula for the derivative of a quotient,

If or

At the point, the derivative changes sign from "minus" to "plus", - the minimum point of the function.

At the point, the derivative changes sign from "plus" to "minus", - the maximum point of the function.

Let's find the values ​​of the function at x=2 and at x=-2.

It is convenient to build function graphs according to a certain algorithm, or scheme. Remember you studied it in school?

The general scheme for constructing a graph of a function:

1. Function scope

2. Range of function values

3. Even - odd (if any)

4. Frequency (if any)

5. Zeros of the function (points where the graph crosses the coordinate axes)

6. Intervals of constancy of a function (that is, intervals on which it is strictly positive or strictly negative).

7. Asymptotes (if any).

8. Behavior of a function at infinity

9. Derivative of a function

10. Intervals of increase and decrease. High and low points and values ​​at these points.