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 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.
The coordinate of absolutely any point on the plane is determined by its two values: along the abscissa axis and the ordinate axis. The set of many such points is the graph of the function. According to it, you can see how the value of Y changes depending on the change in the value of X. You can also determine in which section (interval) the function increases and in which it decreases.
Instruction
- What can be said about a function if its graph is a straight line? See if this line passes through the origin of the coordinates (that is, the one where the X and Y values are 0). If it passes, then such a function is described by the equation y = kx. It is easy to understand that the greater the value of k, the closer this line will be to the y-axis. And the Y-axis itself actually corresponds to an infinitely large value of k.
- Look at the direction of the function. If it goes “bottom left - top right”, that is, through the 3rd and 1st coordinate quarters, it is increasing, but if “top left - right down” (through the 2nd and 4th quarters), then it decreasing.
- When the line does not pass through the origin, it is described by the equation y = kx + b. The line intersects the y-axis at the point where y = b, and the value of y can be either positive or negative.
- A function is called a parabola if it is described by the equation y = x^n, and its form depends on the value of n. If n is any even number (the simplest case is a quadratic function y = x^2), the graph of the function is a curve passing through the origin point, as well as through points with coordinates (1; 1), (-1; 1), for a unit to any power will remain a unit. All y-values corresponding to any non-zero X-values can only be positive. The function is symmetrical about the Y axis, and its graph is located in the 1st and 2nd coordinate quarters. It can be easily understood that the larger the value of n, the closer the graph will be to the Y axis.
- If n is an odd number, the graph of this function is a cubic parabola. The curve is located in the 1st and 3rd coordinate quarters, is symmetrical about the Y axis and passes through the origin, as well as through the points (-1;-1), (1;1). When the quadratic function is the equation y = ax^2 + bx + c, the shape of the parabola is the same as in the simplest case (y = x^2), but its vertex is not at the origin.
- A function is called a hyperbola if it is described by the equation y = k/x. It can be easily seen that as the value of x tends to 0, the value of y increases to infinity. The function graph is a curve consisting of two branches and located in different coordinate quarters.
This methodological material is for reference only and covers a wide range of topics. The article provides an overview of the graphs of the main elementary functions and considers the most important issue - how to correctly and FAST build a graph. In the course of studying higher mathematics without knowing the graphs of the basic elementary functions, it will be difficult, so it is very important to remember what the graphs of a parabola, hyperbola, sine, cosine, etc. look like, to remember some function values. We will also talk about some properties of the main functions.
I do not pretend to completeness and scientific thoroughness of the materials, the emphasis will be placed, first of all, on practice - those things with which one has to face literally at every step, in any topic of higher mathematics. Charts for dummies? You can say so.
By popular demand from readers clickable table of contents:
In addition, there is an ultra-short abstract on the topic
– master 16 types of charts by studying SIX pages!
Seriously, six, even I myself was surprised. This abstract contains improved graphics and is available for a nominal fee, a demo version can be viewed. It is convenient to print the file so that the graphs are always at hand. Thanks for supporting the project!
And we start right away:
How to build coordinate axes correctly?
In practice, tests are almost always drawn up by students in separate notebooks, lined in a cage. Why do you need checkered markings? After all, the work, in principle, can be done on A4 sheets. And the cage is necessary just for the high-quality and accurate design of the drawings.
Any drawing of a function graph starts with coordinate axes.
Drawings are two-dimensional and three-dimensional.
Let us first consider the two-dimensional case Cartesian coordinate system:
1) We draw coordinate axes. The axis is called x-axis , and the axis y-axis . We always try to draw them neat and not crooked. The arrows should also not resemble Papa Carlo's beard.
2) We sign the axes with capital letters "x" and "y". Don't forget to sign the axes.
3) Set the scale along the axes: draw zero and two ones. When making a drawing, the most convenient and common scale is: 1 unit = 2 cells (drawing on the left) - stick to it if possible. However, from time to time it happens that the drawing does not fit on a notebook sheet - then we reduce the scale: 1 unit = 1 cell (drawing on the right). Rarely, but it happens that the scale of the drawing has to be reduced (or increased) even more
DO NOT scribble from a machine gun ... -5, -4, -3, -1, 0, 1, 2, 3, 4, 5, .... For the coordinate plane is not a monument to Descartes, and the student is not a dove. We put zero and two units along the axes. Sometimes instead of units, it is convenient to “detect” other values, for example, “two” on the abscissa axis and “three” on the ordinate axis - and this system (0, 2 and 3) will also uniquely set the coordinate grid.
It is better to estimate the estimated dimensions of the drawing BEFORE the drawing is drawn.. So, for example, if the task requires drawing a triangle with vertices , , , then it is quite clear that the popular scale 1 unit = 2 cells will not work. Why? Let's look at the point - here you have to measure fifteen centimeters down, and, obviously, the drawing will not fit (or barely fit) on a notebook sheet. Therefore, we immediately select a smaller scale 1 unit = 1 cell.
By the way, about centimeters and notebook cells. Is it true that there are 15 centimeters in 30 notebook cells? Measure in a notebook for interest 15 centimeters with a ruler. In the USSR, perhaps this was true ... It is interesting to note that if you measure these same centimeters horizontally and vertically, then the results (in cells) will be different! Strictly speaking, modern notebooks are not checkered, but rectangular. It may seem like nonsense, but drawing, for example, a circle with a compass in such situations is very inconvenient. To be honest, at such moments you begin to think about the correctness of Comrade Stalin, who was sent to camps for hack work in production, not to mention the domestic automotive industry, falling planes or exploding power plants.
Speaking of quality, or a brief recommendation on stationery. To date, most of the notebooks on sale, without saying bad words, are complete goblin. For the reason that they get wet, and not only from gel pens, but also from ballpoint pens! Save on paper. For the design of tests, I recommend using the notebooks of the Arkhangelsk Pulp and Paper Mill (18 sheets, cell) or Pyaterochka, although it is more expensive. It is advisable to choose a gel pen, even the cheapest Chinese gel refill is much better than a ballpoint pen, which either smears or tears paper. The only "competitive" ballpoint pen in my memory is the Erich Krause. She writes clearly, beautifully and stably - either with a full stem, or with an almost empty one.
Additionally: the vision of a rectangular coordinate system through the eyes of analytical geometry is covered in the article Linear (non) dependence of vectors. Vector basis, detailed information about coordinate quarters can be found in the second paragraph of the lesson Linear inequalities.
3D case
It's almost the same here.
1) We draw coordinate axes. Standard: applicate axis – directed upwards, axis – directed to the right, axis – downwards to the left strictly at an angle of 45 degrees.
2) We sign the axes.
3) Set the scale along the axes. Scale along the axis - two times smaller than the scale along the other axes. Also note that in the right drawing, I used a non-standard "serif" along the axis (this possibility has already been mentioned above). From my point of view, it’s more accurate, faster and more aesthetically pleasing - you don’t need to look for the middle of the cell under a microscope and “sculpt” the unit right up to the origin.
When doing a 3D drawing again - give priority to scale
1 unit = 2 cells (drawing on the left).
What are all these rules for? Rules are there to be broken. What am I going to do now. The fact is that the subsequent drawings of the article will be made by me in Excel, and the coordinate axes will look incorrect in terms of proper design. I could draw all the graphs by hand, but it’s really scary to draw them, as Excel is reluctant to draw them much more accurately.
Graphs and basic properties of elementary functions
The linear function is given by the equation . Linear function graph is direct. In order to construct a straight line, it is enough to know two points.
Example 1
Plot the function. Let's find two points. It is advantageous to choose zero as one of the points.
If , then
We take some other point, for example, 1.
If , then
When preparing tasks, the coordinates of points are usually summarized in a table:
And the values themselves are calculated orally or on a draft, calculator.
Two points are found, let's draw:
When drawing up a drawing, we always sign the graphics.
It will not be superfluous to recall special cases of a linear function:
Notice how I placed the captions, signatures should not be ambiguous when studying the drawing. In this case, it was highly undesirable to put a signature next to the point of intersection of the lines, or at the bottom right between the graphs.
1) A linear function of the form () is called direct proportionality. For example, . The direct proportionality graph always passes through the origin. Thus, the construction of a straight line is simplified - it is enough to find only one point.
2) An equation of the form defines a straight line parallel to the axis, in particular, the axis itself is given by the equation. The graph of the function is built immediately, without finding any points. That is, the entry should be understood as follows: "y is always equal to -4, for any value of x."
3) An equation of the form defines a straight line parallel to the axis, in particular, the axis itself is given by the equation. The graph of the function is also built immediately. The entry should be understood as follows: "x is always, for any value of y, equal to 1."
Some will ask, well, why remember the 6th grade?! That's how it is, maybe so, only during the years of practice I met a good dozen students who were baffled by the task of constructing a graph like or .
Drawing a straight line is the most common action when making drawings.
The straight line is discussed in detail in the course of analytic geometry, and those who wish can refer to the article Equation of a straight line on a plane.
Quadratic function graph, cubic function graph, polynomial graph
Parabola. Graph of a quadratic function 
() is a parabola. Consider the famous case:
Let's recall some properties of the function.
So, the solution to our equation: - it is at this point that the vertex of the parabola is located. Why this is so can be learned from the theoretical article on the derivative and the lesson on the extrema of the function. In the meantime, we calculate the corresponding value of "y":
So the vertex is at the point
Now we find other points, while brazenly using the symmetry of the parabola. It should be noted that the function 
– is not even, but, nevertheless, no one canceled the symmetry of the parabola.
In what order to find the remaining points, I think it will be clear from the final table:
This construction algorithm can be figuratively called a "shuttle" or the "back and forth" principle with Anfisa Chekhova.
Let's make a drawing:
From the considered graphs, another useful feature comes to mind:
For a quadratic function 
() the following is true:
If , then the branches of the parabola are directed upwards.
If , then the branches of the parabola are directed downwards.
In-depth knowledge of the curve can be obtained in the lesson Hyperbola and parabola.
The cubic parabola is given by the function . Here is a drawing familiar from school:
We list the main properties of the function
Function Graph
It represents one of the branches of the parabola. Let's make a drawing:
The main properties of the function:
In this case, the axis is vertical asymptote for the hyperbola graph at .
It will be a BIG mistake if, when drawing up a drawing, by negligence, you allow the graph to intersect with the asymptote.
Also one-sided limits, tell us that a hyperbole not limited from above and not limited from below.
Let's explore the function at infinity: , that is, if we start to move along the axis to the left (or right) to infinity, then the “games” will be a slender step infinitely close approach zero, and, accordingly, the branches of the hyperbola infinitely close approach the axis.
So the axis is horizontal asymptote for the graph of the function, if "x" tends to plus or minus infinity.
The function is odd, which means that the hyperbola is symmetrical with respect to the origin. This fact is obvious from the drawing, in addition, it can be easily verified analytically: 
.
The graph of a function of the form () represents two branches of a hyperbola.
If , then the hyperbola is located in the first and third coordinate quadrants(see picture above).
If , then the hyperbola is located in the second and fourth coordinate quadrants.
It is not difficult to analyze the specified regularity of the place of residence of the hyperbola from the point of view of geometric transformations of graphs.
Example 3
Construct the right branch of the hyperbola
We use the pointwise construction method, while it is advantageous to select the values so that they divide completely:
Let's make a drawing:
It will not be difficult to construct the left branch of the hyperbola, here the oddness of the function will just help. Roughly speaking, in the pointwise construction table, mentally add a minus to each number, put the corresponding dots and draw the second branch.
Detailed geometric information about the considered line can be found in the article Hyperbola and parabola.
Graph of an exponential function
In this paragraph, I will immediately consider the exponential function, since in problems of higher mathematics in 95% of cases it is the exponent that occurs.
I remind you that - this is an irrational number: , this will be required when building a graph, which, in fact, I will build without ceremony. Three points is probably enough:
Let's leave the graph of the function alone for now, about it later.
The main properties of the function:
Fundamentally, the graphs of functions look the same, etc.
I must say that the second case is less common in practice, but it does occur, so I felt it necessary to include it in this article.
Graph of a logarithmic function
Consider a function with natural logarithm .
Let's do a line drawing:
If you forgot what a logarithm is, please refer to school textbooks.
The main properties of the function:
Domain: 
Range of values: .
The function is not limited from above: 
, albeit slowly, but the branch of the logarithm goes up to infinity.
Let us examine the behavior of the function near zero on the right: 
. So the axis is vertical asymptote
for the graph of the function with "x" tending to zero on the right.
Be sure to know and remember the typical value of the logarithm: .
Fundamentally, the plot of the logarithm at the base looks the same: , , (decimal logarithm to base 10), etc. At the same time, the larger the base, the flatter the chart will be.
We will not consider the case, something I don’t remember when the last time I built a graph with such a basis. Yes, and the logarithm seems to be a very rare guest in problems of higher mathematics.
In conclusion of the paragraph, I will say one more fact: Exponential Function and Logarithmic Functionare two mutually inverse functions. If you look closely at the graph of the logarithm, you can see that this is the same exponent, just it is located a little differently.
Graphs of trigonometric functions
How does trigonometric torment begin at school? Correctly. From the sine
Let's plot the function
This line is called sinusoid.
I remind you that “pi” is an irrational number:, and in trigonometry it dazzles in the eyes.
The main properties of the function:
This function is periodical with a period. What does it mean? Let's look at the cut. To the left and to the right of it, exactly the same piece of the graph repeats endlessly.
Domain: , that is, for any value of "x" there is a sine value.
Range of values: . The function is limited: , that is, all the “games” sit strictly in the segment .
This does not happen: or, more precisely, it happens, but these equations do not have a solution.
a function is a correspondence between elements of two sets, established according to such a rule that each element of one set is associated with some element from another set.
the graph of a function is the locus of points in the plane whose abscissas (x) and ordinates (y) are connected by the specified function:
the point is located (or is located) on the graph of the function if and only if .
Thus, a function can be adequately described by its graph.
tabular way. Quite common, it consists in setting a table of individual argument values and their corresponding function values. This method of defining a function is used when the domain of the function is a discrete finite set.
With the tabular method of defining a function, it is possible to approximately calculate the values of the function that are not contained in the table, corresponding to the intermediate values of the argument. To do this, use the method of interpolation.
The advantages of the tabular way of specifying a function are that it makes it possible to determine certain specific values at once, without additional measurements or calculations. However, in some cases, the table does not define the function completely, but only for some values of the argument and does not provide a visual representation of the nature of the change in the function depending on the change in the argument.
Graphic way. The graph of the function y = f(x) is the set of all points in the plane whose coordinates satisfy the given equation.
The graphical way of specifying a function does not always make it possible to accurately determine the numerical values of the argument. However, it has a great advantage over other methods - visibility. In engineering and physics, a graphical method of setting a function is often used, and a graph is the only way available for this.
In order for the graphical assignment of a function to be quite correct from a mathematical point of view, it is necessary to indicate the exact geometric construction of the graph, which, most often, is given by an equation. This leads to the following way of defining a function.
analytical way. Most often, the law that establishes a relationship between an argument and a function is specified by means of formulas. This way of defining a function is called analytical.
This method makes it possible for each numerical value of the argument x to find the corresponding numerical value of the function y exactly or with some accuracy.
If the relationship between x and y is given by a formula that is resolved with respect to y, i.e. has the form y = f(x), then we say that the function of x is given explicitly.
If the values x and y are related by some equation of the form F(x,y) = 0, i.e. the formula is not allowed with respect to y, which means that the function y = f(x) is implicitly defined.
A function can be defined by different formulas in different parts of its task area.
The analytical method is the most common way to define functions. Compactness, conciseness, the ability to calculate the value of a function for an arbitrary value of the argument from the domain of definition, the ability to apply the apparatus of mathematical analysis to a given function are the main advantages of the analytical method of defining a function. The disadvantages include the lack of visibility, which is compensated by the ability to build a graph and the need to perform sometimes very cumbersome calculations.
verbal way. This method consists in the fact that the functional dependence is expressed in words.
Example 1: the function E(x) is the integer part of the number x. In general, E(x) = [x] denotes the largest integer that does not exceed x. In other words, if x = r + q, where r is an integer (may be negative) and q belongs to the interval = r. The function E(x) = [x] is constant on the interval = r.
Example 2: function y = (x) - fractional part of a number. More precisely, y =(x) = x - [x], where [x] is the integer part of the number x. This function is defined for all x. If x is an arbitrary number, then representing it as x = r + q (r = [x]), where r is an integer and q lies in the interval .
We see that adding n to the x argument does not change the value of the function.
The smallest non-zero number in n is , thus the period is sin 2x .
The value of the argument for which the function is equal to 0 is called zero (root) functions.
A function can have multiple zeros.
For example, the function y=x(x+1)(x-3) has three zeros: x=0, x=-1, x=3.
Geometrically, the zero of a function is the abscissa of the intersection point of the graph of the function with the axis X .
Figure 7 shows the graph of the function with zeros: x = a, x = b and x = c .
If the graph of a function approaches a certain straight line indefinitely as it moves away from the origin, then this straight line is called asymptote.
Inverse function
Let the function y=ƒ(x) be given with the domain of definition D and the set of values E. If each value yєE corresponds to a single value xєD, then the function x=φ(y) is defined with the domain of definition E and the set of values D (see Fig. 102 ).
Such a function φ(y) is called the inverse of the function ƒ(x) and is written in the following form: x=j(y)=f -1 (y). About the functions y=ƒ(x) and x=φ(y) they say that they are mutually inverse. To find the function x=φ(y) inverse to the function y=ƒ(x), it is sufficient to solve the equation ƒ(x)=y with respect to x (if possible).
1. For the function y \u003d 2x, the inverse function is the function x \u003d y / 2;
2. For the function y \u003d x2 xє, the inverse function is x \u003d √y; note that for the function y \u003d x 2, given on the segment [-1; 1], there is no inverse, since one value of y corresponds to two values of x (for example, if y=1/4, then x1=1/2, x2=-1/2).
It follows from the definition of the inverse function that the function y=ƒ(x) has an inverse if and only if the function ƒ(x) defines a one-to-one correspondence between the sets D and E. It follows that any strictly monotonic function has an inverse. Moreover, if the function increases (decreases), then the inverse function also increases (decreases).
Note that the function y \u003d ƒ (x) and its inverse x \u003d φ (y) are depicted by the same curve, that is, their graphs coincide. If we agree that, as usual, the independent variable (i.e., the argument) is denoted by x, and the dependent variable by y, then the inverse function of the function y \u003d ƒ (x) will be written as y \u003d φ (x).
This means that the point M 1 (x o; y o) of the curve y=ƒ(x) becomes the point M 2 (y o; x o) of the curve y=φ(x). But the points M 1 and M 2 are symmetrical about the straight line y \u003d x (see Fig. 103). Therefore, the graphs of mutually inverse functions y=ƒ(x) and y=φ(x) are symmetrical with respect to the bisector of the first and third coordinate angles.
Complex function
Let the function y=ƒ(u) be defined on the set D, and the function u= φ(х) on the set D 1 , and for x D 1 the corresponding value u=φ(x) є D. Then on the set D 1 is defined function u=ƒ(φ(x)), which is called a complex function of x (or a superposition of given functions, or a function of a function).
The variable u=φ(x) is called an intermediate argument of a complex function.
For example, the function y=sin2x is a superposition of two functions y=sinu and u=2x. A complex function can have multiple intermediate arguments.
4. Basic elementary functions and their graphs.
The following functions are called basic elementary functions.
1) The exponential function y \u003d a x, a> 0, a ≠ 1. In fig. 104 shows graphs of exponential functions corresponding to various exponential bases.
2) Power function y=x α , αєR. Examples of graphs of power functions corresponding to various exponents are provided in the figures
3) Logarithmic function y=log a x, a>0,a≠1; Graphs of logarithmic functions corresponding to different bases are shown in fig. 106.
4) Trigonometric functions y=sinx, y=cosx, y=tgx, y=ctgx; Graphs of trigonometric functions have the form shown in fig. 107.
5) Inverse trigonometric functions y=arcsinx, y=arccosx, y=arctgx, y=arcctgx. On fig. 108 shows graphs of inverse trigonometric functions.
A function given by one formula, composed of basic elementary functions and constants using a finite number of arithmetic operations (addition, subtraction, multiplication, division) and operations of taking a function from a function, is called an elementary function.
Examples of elementary functions are the functions
Examples of non-elementary functions are the functions
5. Concepts of the limit of a sequence and a function. Limit properties.
Function limit (function limit) at a given point, limiting for the domain of definition of a function, is such a value to which the value of the function under consideration tends when its argument tends to a given point.
In mathematics sequence limit elements of a metric space or a topological space is an element of the same space that has the property of "attracting" elements of a given sequence. The limit of a sequence of elements of a topological space is such a point, each neighborhood of which contains all the elements of the sequence, starting from some number. In a metric space, neighborhoods are defined in terms of a distance function, so the concept of a limit is formulated in the language of distances. Historically, the first was the concept of the limit of a numerical sequence, which arises in mathematical analysis, where it serves as the basis for a system of approximations and is widely used in the construction of differential and integral calculus.
Designation:
(read: the limit of the x-nth sequence as en tending to infinity is a)
The property of a sequence to have a limit is called convergence: if a sequence has a limit, then the given sequence is said to be converges; otherwise (if the sequence has no limit) the sequence is said to be diverges. In a Hausdorff space, and in particular a metric space, every subsequence of a convergent sequence converges, and its limit is the same as the limit of the original sequence. In other words, a sequence of elements in a Hausdorff space cannot have two different limits. It may, however, turn out that the sequence has no limit, but there is a subsequence (of the given sequence) that has a limit. If any sequence of points in a space has a convergent subsequence, then the given space is said to have the property of sequential compactness (or, simply, compactness if compactness is defined exclusively in terms of sequences).
The concept of the limit of a sequence is directly related to the concept of a limit point (set): if a set has a limit point, then there is a sequence of elements of the given set converging to the given point.
Definition
Let a topological space and a sequence be given Then, if there exists an element such that
where is an open set containing , then it is called the limit of the sequence . If the space is metric, then the limit can be defined using a metric: if there exists an element such that
where is the metric, then is called the limit.
· If a space is equipped with an antidiscrete topology, then the limit of any sequence is any element of the space.
6. Limit of a function at a point. Unilateral limits.
Function of one variable. Determining the limit of a function at a point according to Cauchy. Number b is called the limit of the function at = f(x) at X striving for a(or at the point a) if for any positive number there is a positive number such that for all x ≠ a, such that | x – a | < , выполняется неравенство
| f(x) – a | < .
Determining the limit of a function at a point according to Heine. Number b is called the limit of the function at = f(x) at X striving for a(or at the point a) if for any sequence ( x n ) converging to a(aspiring to a, which has a limit number a), and for any value n x n≠ a, subsequence ( y n= f(x n)) converges to b.
These definitions assume that the function at = f(x) is defined in some neighborhood of the point a, except perhaps for the very point a.
The definitions of the limit of a function at a point according to Cauchy and according to Heine are equivalent: if the number b serves as a limit in one of them, then the same is true in the second.
The specified limit is indicated as follows:
Geometrically, the existence of the limit of a function at a point according to Cauchy means that for any number > 0, such a rectangle can be indicated on the coordinate plane with a base 2 > 0, a height 2 and a center at the point ( a; b) that all points of the graph of this function on the interval ( a– ; a+ ), with the possible exception of the point M(a; f(a)), lie in this rectangle
One-sided limit in mathematical analysis, the limit of a numerical function, implying "approaching" the limit point from one side. Such limits are called respectively left-hand limit(or left limit) and right-hand limit (limit on the right). Let a numerical function be given on some numerical set and the number be the limit point of the domain of definition. There are various definitions for the one-sided limits of a function at a point, but they are all equivalent.
National Research University
Department of Applied Geology
Essay on higher mathematics
On the topic: "Basic elementary functions,
their properties and graphs"
Completed:
Checked:
teacher
Definition. The function given by the formula y=a x (where a>0, a≠1) is called an exponential function with base a.
Let us formulate the main properties of the exponential function:
1. The domain of definition is the set (R) of all real numbers.
2. The range of values is the set (R+) of all positive real numbers.
3. When a > 1, the function increases on the entire real line; at 0<а<1 функция убывает.
4. Is a general function.
, on the interval xн [-3;3]
, on the interval xн [-3;3] A function of the form y(х)=х n , where n is the number ОR, is called a power function. The number n can take on different values: both integer and fractional, both even and odd. Depending on this, the power function will have a different form. Consider special cases that are power functions and reflect the main properties of this type of curves in the following order: power function y \u003d x² (a function with an even exponent - a parabola), a power function y \u003d x³ (a function with an odd exponent - a cubic parabola) and function y \u003d √ x (x to the power of ½) (function with a fractional exponent), a function with a negative integer exponent (hyperbola).
Power function y=x²
1. D(x)=R – the function is defined on the entire numerical axis;
2. E(y)= and increases on the interval
Power function y=x³
1. The graph of the function y \u003d x³ is called a cubic parabola. The power function y=x³ has the following properties:
2. D(x)=R – the function is defined on the entire numerical axis;
3. E(y)=(-∞;∞) – the function takes all values in its domain of definition;
4. When x=0 y=0 – the function passes through the origin O(0;0).
5. The function increases over the entire domain of definition.
6. The function is odd (symmetric about the origin).
, on the interval xн [-3;3] Depending on the numerical factor in front of x³, the function can be steep / flat and increase / decrease.
Power function with integer negative exponent:
If the exponent n is odd, then the graph of such a power function is called a hyperbola. A power function with a negative integer exponent has the following properties:
1. D(x)=(-∞;0)U(0;∞) for any n;
2. E(y)=(-∞;0)U(0;∞) if n is an odd number; E(y)=(0;∞) if n is an even number;
3. The function decreases over the entire domain of definition if n is an odd number; the function increases on the interval (-∞;0) and decreases on the interval (0;∞) if n is an even number.
4. The function is odd (symmetric about the origin) if n is an odd number; a function is even if n is an even number.
5. The function passes through the points (1;1) and (-1;-1) if n is an odd number and through the points (1;1) and (-1;1) if n is an even number.
, on the interval xн [-3;3] Power function with fractional exponent
A power function with a fractional exponent of the form (picture) has a graph of the function shown in the figure. A power function with a fractional exponent has the following properties: (picture)
1. D(x) нR if n is an odd number and D(x)= 
, on the interval xн 
, on the interval xн [-3;3]
The logarithmic function y \u003d log a x has the following properties:
1. Domain of definition D(x)н (0; + ∞).
2. Range of values E(y) О (- ∞; + ∞)
3. The function is neither even nor odd (general).
4. The function increases on the interval (0; + ∞) for a > 1, decreases on (0; + ∞) for 0< а < 1.
The graph of the function y = log a x can be obtained from the graph of the function y = a x using a symmetry transformation about the line y = x. In Figure 9, a plot of the logarithmic function for a > 1 is plotted, and in Figure 10 - for 0< a < 1.
; on the interval xО 
; on the interval xО The functions y \u003d sin x, y \u003d cos x, y \u003d tg x, y \u003d ctg x are called trigonometric functions.
The functions y \u003d sin x, y \u003d tg x, y \u003d ctg x are odd, and the function y \u003d cos x is even.
Function y \u003d sin (x).
1. Domain of definition D(x) ОR.
2. Range of values E(y) О [ - 1; one].
3. The function is periodic; the main period is 2π.
4. The function is odd.
5. The function increases on the intervals [ -π/2 + 2πn; π/2 + 2πn] and decreases on the intervals [ π/2 + 2πn; 3π/2 + 2πn], n О Z.
The graph of the function y \u003d sin (x) is shown in Figure 11.
