Theory of elementary functions. Basic elementary functions

Knowledge basic elementary functions, their properties and graphs no less important than knowing the multiplication tables. They are like the 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 main elementary functions, provide their graphs and give without conclusion or proof properties of basic elementary functions according to the scheme:

• behavior of a function at the boundaries of the domain of definition, vertical asymptotes (if necessary, see the article classification of discontinuity points 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, conditions of convexity and inflection);
• oblique and horizontal asymptotes;
• singular points of functions;
• special properties of some functions (for example, the smallest positive period of 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), nth 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. A constant function associates each real value of the independent variable x with the same value of the dependent variable y - the value C. 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 the 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: the entire set of real numbers.
• The constant function is even.
• Range of values: a set consisting of the singular number C.
• A constant function is non-increasing and non-decreasing (that’s why it’s constant).
• It makes no sense to talk about convexity and concavity of a constant.
• There are no asymptotes.
• The function passes through the point (0,C) of the coordinate plane.

nth root.

Let's consider the basic elementary function, which is given by the formula , where n is a natural number greater than one.

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.

As an example, here is a picture with images of function graphs and , they correspond to black, red and blue lines.

The graphs of even-degree root functions have a similar appearance for other values ​​of the exponent.

Properties of the nth root function for even n.

The nth root, n is an odd number.

The nth root function with an odd root exponent n is defined on the entire set of real numbers. For example, here are the function graphs and , they correspond to black, red and blue curves.

For other odd values ​​of the root exponent, the function graphs 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 .

Let's consider the form 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 appearance of the 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 will 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 type of graphs of such power functions) depend on the value of the exponent a. We will consider them, firstly, for a from zero to one, secondly, for a greater than one, thirdly, for a from minus one to zero, fourthly, for a less than minus one.

At the end of this section, for completeness, we will describe a power function with zero exponent.

Power function with odd positive exponent.

Let's 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.

Let's consider a power function with an even positive exponent, that is, for a = 2,4,6,....

As an example, we give graphs of power functions – black line, – blue line, – red line. For a=2 we have a quadratic function, the graph of which is quadratic parabola.

Properties of a power function with an even positive exponent.

Power function with 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 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 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.

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 domain of definition of the power function to be the interval. It is stipulated that the exponent a is an irreducible fraction. Now the authors of many textbooks on algebra and principles 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 precisely this view, that is, we will consider the set to be the domains of definition of power functions with fractional positive exponents. We recommend that students find out your teacher's opinion on this subtle point in order to avoid disagreements.

Let us consider a power function with a rational or irrational exponent a, and .

Let us 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.

Let us consider a power function with a non-integer rational or irrational exponent a, and .

Let us present graphs of 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 appearance.

Properties of the power function at .

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 domain of definition of a power function to be the interval . It is stipulated that the exponent a is an irreducible fraction. Now the authors of many textbooks on algebra and principles 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 precisely this view, that is, we will consider the domains of definition of power functions with fractional fractional negative exponents to be a set, respectively. We recommend that students find out your teacher's opinion on this subtle point in order to avoid disagreements.

Let's move on to the power function, kgod.

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 that is less than minus one.

Let us give examples of graphs of power functions for , they are depicted by 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, we have a function - this is a straight line from which the point (0;1) is excluded (it was agreed not to attach any significance to the expression 0 0).

Exponential function.

One of the main elementary functions is the exponential function.

The graph of the exponential function, where and takes different forms depending on the value of the base a. Let's figure this out.

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

As an example, we present graphs of the exponential function for a = 1/2 – blue line, a = 5/6 – 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.

Let us move on 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 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 a logarithmic function takes different forms depending on the value of the base a.

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. Exponential function $y=a^(x) $, where the base is degree $a>0$, $a\ne 1$.
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

We will consider the behavior of the power function $y=x^(a) $ for those 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, that is, $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 $\left(x\to +\infty \ right)$, and with 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 $, which mean that the function in both cases increases without limit ($\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, that is, $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 with unlimited increase 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 zero. In addition, as the argument tends to zero both on the left $\left(x\to 0-0\right)$ and on the right $\left(x\to 0+0\right)$, the function increases without limit. Therefore, the expressions $\mathop(\lim )\limits_(x\to 0-0) \frac(1)(x^(2\cdot k) ) =+\infty $ and $\mathop(\lim )\limits_ are valid (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 both when the argument increases and when it decreases without limit. 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 on the left, the function decreases without limit, and as the argument approaches zero on the right, the function increases without limit, 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 inverse of the natural number, that is, $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 $x\ge 0$. With an unlimited increase in the argument, the value of the function $y=+\sqrt[(2\cdot k)](x) $ increases unlimitedly, and the value of the function $y=-\sqrt[(2\cdot k)](x) $ decreases unlimitedly , that is, $\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 unlimitedly with an unlimited increase in the argument and decreases unlimitedly when unlimited, it decreases, that is, $\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 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.

When the argument $\left(x\to +\infty \right)$ increases indefinitely, the exponential function or $\mathop(\lim )\limits_(x\to +\infty ) a^(x) =+\infty $ increases indefinitely , if $a>1$, or asymptotically approaches zero $\mathop(\lim )\limits_(x\to +\infty ) a^(x) =0$, if $a1$, or $\mathop increases without limit (\lim )\limits_(x\to -\infty ) a^(x) =+\infty $, if $a

The characteristic value for the function $y=a^(x) $ is the value $x=0$. In this case, 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 $\mathop(\lim )\limits_(x\to +\infty ) \log _(a) x=+\ increases indefinitely infty $, if $a>1$, or decreases without limit $\mathop(\lim )\limits_(x\to +\infty ) \log _(a) x=-\infty $, if $a1$, or without limit $\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 $y=0$. In this case, all logarithmic functions, regardless of $a$, necessarily intersect the $Ox$ axis at $x=1$. Examples: graphs of the 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 decimal, 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.

The section contains reference material on the main elementary functions and their properties. A classification of elementary functions is given. Below are links to subsections that discuss the properties of specific functions - graphs, formulas, derivatives, antiderivatives (integrals), series expansions, expressions through complex variables.

Reference pages for basic functions

Classification of elementary functions

Algebraic function is a function that satisfies the equation:
,
where is a polynomial in the dependent variable y and the independent variable x. It can be written as:
,
where are polynomials.

Algebraic functions are divided into polynomials (entire rational functions), rational functions and irrational functions.

Entire rational function, which is also called polynomial or polynomial, is obtained from the variable x and a finite number of numbers using the arithmetic operations of addition (subtraction) and multiplication. After opening the brackets, the polynomial is reduced to canonical form:
.

Fractional rational function, or simply rational function, is obtained from the variable x and a finite number of numbers using the arithmetic operations of addition (subtraction), multiplication and division. The rational function can be reduced to the form
,
where and are polynomials.

Irrational function is an algebraic function that is not rational. As a rule, an irrational function is understood as roots and their compositions with rational functions. A root of degree n is defined as the solution to the equation
.
It is designated as follows:
.

Transcendental functions are called non-algebraic functions. These are exponential, trigonometric, hyperbolic and their inverse functions.

Overview of basic elementary functions

All elementary functions can be represented as a finite number of addition, subtraction, multiplication and division operations performed on an expression of the form:
z t .
Inverse functions can also be expressed in terms of logarithms. The basic elementary functions are listed below.

Power function :
y(x) = x p ,
where p is the exponent. It depends on the base of the degree x.
The inverse of the power function is also the power function:
.
For an integer non-negative value of the exponent p, it is a polynomial. For an integer value p - a rational function. With a rational meaning - an irrational function.

Transcendental functions

Exponential function :
y(x) = a x ,
where a is the base of the degree. It depends on the exponent x.
The inverse function is the logarithm to base a:
x = log a y.

Exponent, e to the x power:
y(x) = e x ,
This is an exponential function whose derivative is equal to the function itself:
.
The base of the exponent is the number e:
≈ 2,718281828459045... .
Inverse function - natural logarithm - logarithm to base e:
x = ln y ≡ log e y.

Trigonometric functions:
Sine: ;
Cosine: ;
Tangent: ;
Cotangent: ;
Here i is the imaginary unit, i 2 = -1.

Inverse trigonometric functions:
Arcsine: x = arcsin y, ;
Arc cosine: x = arccos y, ;
Arctangent: x = arctan y, ;
Arc tangent: x = arcctg y, .

Basic elementary functions are: constant function (constant), root n-th degree, power function, exponential, logarithmic function, trigonometric and inverse trigonometric functions.

Permanent function.

A constant function is given on the set of all real numbers by the formula , where C– some real number. A constant function assigns each actual value of the independent variable x same value of the dependent variable y- meaning WITH. 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 the 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: the entire set of real numbers.

The constant function is even.

Range of values: set consisting of a singular number WITH.

A constant function is non-increasing and non-decreasing (that’s why it’s constant).

It makes no sense to talk about convexity and concavity of a constant.

There are no asymptotes.

The function passes through the point (0,C) coordinate plane.

Root of the nth degree.

Let's consider the basic elementary function, which is given by the formula, where n– a natural number greater than one.

The nth root, n is an even number.

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

As an example, here is a picture with images of function graphs and , they correspond to black, red and blue lines.

The graphs of even-degree root functions have a similar appearance for other values ​​of the exponent.

Properties of the root functionn -th power for evenn .

The nth root, n is an odd number.

Root function n-th power with an odd root exponent n is defined on the entire set of real numbers. For example, here are the function graphs and , they correspond to black, red and blue curves.