Basic elementary functions: their properties and graphs. Basic properties of functions What is an elementary function

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

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

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

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

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

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Permanent function.

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

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

Properties of a constant function.

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

The root of the nth degree.

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

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

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

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

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

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

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

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

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

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

Power function.

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

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

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

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

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

Power function with odd positive exponent.

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

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

Properties of a power function with an odd positive exponent.

Power function with even positive exponent.

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

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

Properties of a power function with an even positive exponent.

Power function with an odd negative exponent.

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

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

Properties of a power function with an odd negative exponent.

Power function with an even negative exponent.

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

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

Properties of a power function with an even negative exponent.

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

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

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

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

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

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

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

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

Power function properties for .

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

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

We pass to the power function , where .

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

Properties of a power function with exponent a , .

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

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

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

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

Exponential function.

One of the basic elementary functions is the exponential function.

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

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

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

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

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

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

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

Logarithmic function.

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

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

Considering the functions of a complex variable, Liouville defined elementary functions somewhat more broadly. elementary function y variable x- an analytic function that can be represented as an algebraic function of x and functions , and is the logarithm or exponent of some algebraic function g 1 from x .

For example, sin( x) is an algebraic function of e ix .

Without limiting the generality of the consideration, we can assume that the functions are algebraically independent, that is, if the algebraic equation is satisfied for all x, then all coefficients of the polynomial are equal to zero.

Differentiation of elementary functions

Where z 1 "(z) is equal to or g 1 " / g 1 or z 1 g 1" depending on whether the logarithm z 1 or exponent, etc. In practice, it is convenient to use a table of derivatives.

Integration of elementary functions

The Liouville theorem is the basis for creating algorithms for the symbolic integration of elementary functions, implemented, for example, in

Limit Calculation

Liouville's theory does not extend to the calculation of limits. It is not known whether there is an algorithm that, given the sequence given by the elementary formula, gives an answer, whether it has a limit or not. For example, the question of whether the sequence converges is open.

Literature

• J. Liouville. Mémoire sur l'integration d'une classe de fonctions transcendantes// J. Reine Angew. Math. bd. 13, p. 93-118. (1835)
• J.F. Ritt. Integration in Finite Terms. N.-Y., 1949// http://lib.homelinux.org
• A. G. Khovansky. Topological Galois theory: solvability and unsolvability of equations in final form Ch. 1. M, 2007

Notes

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• Elementary excitement
• Elementary Exodus

See what "Elementary function" is in other dictionaries:

elementary function- A function that, if divided into smaller functions, cannot be uniquely identified in the digital transmission hierarchy. Therefore, from the point of view of the network, it is indivisible (ITU T G.806). Telecommunication topics, basic concepts EN adaptation functionA ... Technical Translator's Handbook

interworking function between network layers- An elementary function that ensures the interaction of the characteristic information between the two levels of the network. (ITU-T G.806). Telecommunication topics, basic concepts of EN layer ... ... Technical Translator's Handbook

Complete list of basic elementary functions

The class of basic elementary functions includes the following:

1. Constant function $y=C$, where $C$ is a constant. Such a function takes the same value $C$ for any $x$.
2. Power function $y=x^(a) $, where the exponent $a$ is a real number.
3. An exponential function $y=a^(x) $, where the base is $a>0$, $a\ne 1$.
4. Logarithmic function $y=\log _(a) x$, where the base of the logarithm is $a>0$, $a\ne 1$.
5. Trigonometric functions $y=\sin x$, $y=\cos x$, $y=tg\, x$, $y=ctg\, x$, $y=\sec x$, $y=A>\ sec \, x$.
6. Inverse trigonometric functions $y=\arcsin x$, $y=\arccos x$, $y=arctgx$, $y=arcctgx$, $y=arc\sec x$, $y=arc\, \cos ec\ , x$.

Power functions

Let us consider the behavior of the power function $y=x^(a) $ for the simplest cases when its exponent determines integer exponentiation and root extraction.

Case 1

The exponent of the function $y=x^(a) $ is a natural number, ie $y=x^(n) $, $n\in N$.

If $n=2\cdot k$ is an even number, then the function $y=x^(2\cdot k) $ is even and increases indefinitely as if the argument $\left(x\to +\infty \ right)$, as well as for its unlimited decrease $\left(x\to -\infty \right)$. This behavior of the function can be described by the expressions $\mathop(\lim )\limits_(x\to +\infty ) x^(2\cdot k) =+\infty $ and $\mathop(\lim )\limits_(x\to -\infty ) x^(2\cdot k) =+\infty $, which means that in both cases the function increases without limit ($\lim $ is the limit). Example: graph of the function $y=x^(2) $.

If $n=2\cdot k-1$ is an odd number, then the function $y=x^(2\cdot k-1) $ is odd, increases indefinitely as the argument increases indefinitely, and decreases indefinitely as the argument decreases indefinitely. This behavior of the function can be described by the expressions $\mathop(\lim )\limits_(x\to +\infty ) x^(2\cdot k-1) =+\infty $ and $\mathop(\lim )\limits_(x \to -\infty ) x^(2\cdot k-1) =-\infty $. Example: graph of the function $y=x^(3) $.

Case 2

The exponent of the function $y=x^(a) $ is a negative integer, i.e. $y=\frac(1)(x^(n) ) $, $n\in N$.

If $n=2\cdot k$ is an even number, then the function $y=\frac(1)(x^(2\cdot k) ) $ is even and asymptotically (gradually) approaches zero as if argument, and with its unlimited decrease. This behavior of the function can be described by a single expression $\mathop(\lim )\limits_(x\to \infty ) \frac(1)(x^(2\cdot k) ) =0$, which means that with an unlimited increase in the argument in absolute value, the limit of the function is equal to zero. Moreover, as the argument tends to zero both from the left $\left(x\to 0-0\right)$ and from the right $\left(x\to 0+0\right)$, the function increases without limit. Therefore, $\mathop(\lim )\limits_(x\to 0-0) \frac(1)(x^(2\cdot k) ) =+\infty $ and $\mathop(\lim )\limits_ (x\to 0+0) \frac(1)(x^(2\cdot k) ) =+\infty $, which means that the function $y=\frac(1)(x^(2\cdot k ) ) $ in both cases has an infinite limit equal to $+\infty $. Example: graph of the function $y=\frac(1)(x^(2) ) $.

If $n=2\cdot k-1$ is an odd number, then the function $y=\frac(1)(x^(2\cdot k-1) ) $ is odd and asymptotically approaches zero as if increase of the argument, and with its unlimited decrease. This behavior of the function can be described by a single expression $\mathop(\lim )\limits_(x\to \infty ) \frac(1)(x^(2\cdot k-1) ) =0$. In addition, as the argument approaches zero from the left, the function decreases indefinitely, and as the argument approaches zero from the right, the function increases indefinitely, that is, $\mathop(\lim )\limits_(x\to 0-0) \frac(1)(x ^(2\cdot k-1) ) =-\infty $ and $\mathop(\lim )\limits_(x\to 0+0) \frac(1)(x^(2\cdot k-1) ) =+\infty$. Example: graph of the function $y=\frac(1)(x) $.

Case 3

The exponent of the function $y=x^(a) $ is the reciprocal of the natural number, i.e. $y=\sqrt[(n)](x) $, $n\in N$.

If $n=2\cdot k$ is an even number, then the function $y=\pm \sqrt[(2\cdot k)](x) $ is two-valued and is defined only for $x\ge 0$. When the argument increases without limit, the value of the function $y=+\sqrt[(2\cdot k)](x) $ increases without limit, and the value of the function $y=-\sqrt[(2\cdot k)](x) $ decreases without limit , i.e. $\mathop(\lim )\limits_(x\to +\infty ) \left(+\sqrt[(2\cdot k)](x) \right)=+\infty $ and $\mathop( \lim )\limits_(x\to +\infty ) \left(-\sqrt[(2\cdot k)](x) \right)=-\infty $. Example: graph of the function $y=\pm \sqrt(x) $.

If $n=2\cdot k-1$ is an odd number, then the function $y=\sqrt[(2\cdot k-1)](x) $ is odd, increases indefinitely as the argument increases indefinitely, and decreases indefinitely when unbounded it decreases, i.e. $\mathop(\lim )\limits_(x\to +\infty ) \sqrt[(2\cdot k-1)](x) =+\infty $ and $\mathop(\ lim )\limits_(x\to -\infty ) \sqrt[(2\cdot k-1)](x)=-\infty $. Example: graph of the function $y=\sqrt[(3)](x) $.

Exponential and logarithmic functions

The exponential $y=a^(x) $ and the logarithmic $y=\log _(a) x$ functions are mutually inverse. Their graphs are symmetrical with respect to the common bisector of the first and third coordinate angles.

As the argument $\left(x\to +\infty \right)$ increases indefinitely, the exponential function or $\mathop(\lim )\limits_(x\to +\infty ) a^(x) =+\infty $ , if $a>1$, or asymptotically approaches zero $\mathop(\lim )\limits_(x\to +\infty ) a^(x) =0$, if $a1$, or $\mathop increases indefinitely (\lim )\limits_(x\to -\infty ) a^(x) =+\infty $ if $a

The characteristic value for the function $y=a^(x) $ is the value $x=0$. Moreover, all exponential functions, regardless of $a$, necessarily intersect the $Oy$ axis at $y=1$. Examples: graphs of functions $y=2^(x) $ and $y = \left (\frac(1)(2) \right)^(x) $.

The logarithmic function $y=\log _(a) x$ is defined only for $x > 0$.

As the argument $\left(x\to +\infty \right)$ increases indefinitely, the logarithmic function or increases indefinitely $\mathop(\lim )\limits_(x\to +\infty ) \log _(a) x=+\ infty $ if $a>1$, or $\mathop(\lim )\limits_(x\to +\infty ) \log _(a) x=-\infty $ if $a1$, or unlimited $\mathop(\lim )\limits_(x\to 0+0) \log _(a) x=+\infty $ increases if $a

The characteristic value for the function $y=\log _(a) x$ is the value $y=0$. Moreover, all logarithmic functions, regardless of $a$, necessarily intersect the $Ox$ axis at $x=1$. Examples: graphs of functions $y=\log _(2) x$ and $y=\log _(1/2) x$.

Some logarithmic functions have special notation. In particular, if the base of the logarithm is $a=10$, then such a logarithm is called a decimal logarithm, and the corresponding function is written as $y=\lg x$. And if the irrational number $e=2.7182818\ldots $ is chosen as the base of the logarithm, then such a logarithm is called natural, and the corresponding function is written as $y=\ln x$. Its inverse is the function $y=e^(x) $, called the exponent.