schedule f. Functions. Basic types, graphs, methods of assignment. A power function with a rational or irrational exponent whose value is greater than zero and less than one

First, try to find the scope of the function:

Did you manage? Let's compare the answers:

All right? Well done!

Now let's try to find the range of the function:

Found? Compare:

Did it agree? Well done!

Let's work with the graphs again, only now it's a little more difficult - to find both the domain of the function and the range of the function.

How to Find Both the Domain and Range of a Function (Advanced)

Here's what happened:

With graphics, I think you figured it out. Now let's try to find the domain of the function in accordance with the formulas (if you don't know how to do this, read the section about):

Did you manage? Checking answers:

, since the root expression must be greater than or equal to zero.
, since it is impossible to divide by zero and the radical expression cannot be negative.
, since, respectively, for all.
because you can't divide by zero.

However, we still have one more moment that has not been sorted out ...

Let me reiterate the definition and focus on it:

Noticed? The word "only" is very, very important element our definition. I will try to explain to you on the fingers.

Let's say we have a function given by a straight line. . At, we substitute given value into our "rule" and we get that. One value corresponds to one value. We can even make a table of various values and plot a given function to verify this.

"Look! - you say, - "" meets twice!" So maybe the parabola is not a function? No, it is!

The fact that "" occurs twice is far from a reason to accuse the parabola of ambiguity!

The fact is that, when calculating for, we got one game. And when calculating with, we got one game. So that's right, the parabola is a function. Look at the chart:

Got it? If not, here's a real-life example for you, far from mathematics!

Let's say we have a group of applicants who met when submitting documents, each of whom told in a conversation where he lives:

Agree, it is quite realistic that several guys live in the same city, but it is impossible for one person to live in several cities at the same time. This is, as it were, a logical representation of our "parabola" - Several different x's correspond to the same y.

Now let's come up with an example where the dependency is not a function. Let's say these same guys told what specialties they applied for:

Here we have a completely different situation: one person can easily apply for one or several directions. That is one element sets are put in correspondence multiple elements sets. Respectively, it's not a function.

Let's test your knowledge in practice.

Determine from the pictures what is a function and what is not:

Got it? And here is answers:

The function is - B,E.
Not a function - A, B, D, D.

You ask why? Yes, here's why:

In all figures except V) and E) there are several for one!

I am sure that now you can easily distinguish a function from a non-function, say what an argument is and what a dependent variable is, and also determine the scope of the argument and the scope of the function. Let's move on to the next section - how to define a function?

Ways to set a function

What do you think the words mean "set function"? That's right, it means explaining to everyone what function we are talking about in this case. Moreover, explain in such a way that everyone understands you correctly and the graphs of functions drawn by people according to your explanation were the same.

How can I do that? How to set a function? The easiest way, which has already been used more than once in this article - using a formula. We write a formula, and by substituting a value into it, we calculate the value. And as you remember, a formula is a law, a rule according to which it becomes clear to us and to another person how an X turns into a Y.

Usually, this is exactly what they do - in tasks we see ready-made functions defined by formulas, however, there are other ways to set a function that everyone forgets about, and therefore the question “how else can you set a function?” confuses. Let's take a look at everything in order, and start with the analytical method.

Analytical way of defining a function

The analytical method is the task of a function using a formula. This is the most universal and comprehensive and unambiguous way. If you have a formula, then you know absolutely everything about the function - you can make a table of values on it, you can build a graph, determine where the function increases and where it decreases, in general, explore it in full.

Let's consider a function. What does it matter?

"What does it mean?" - you ask. I'll explain now.

Let me remind you that in the notation, the expression in brackets is called the argument. And this argument can be any expression, not necessarily simple. Accordingly, whatever the argument (expression in brackets), we will write it instead in the expression.

In our example, it will look like this:

Consider another task related to the analytical method of specifying a function that you will have on the exam.

Find the value of the expression, at.

I'm sure that at first, you were scared when you saw such an expression, but there is absolutely nothing scary in it!

Everything is the same as in the previous example: whatever the argument (expression in brackets), we will write it instead in the expression. For example, for a function.

What should be done in our example? Instead, you need to write, and instead of -:

shorten the resulting expression:

That's all!

Independent work

Now try to find the meaning of the following expressions yourself:

, if
, if

Did you manage? Let's compare our answers: We are used to the fact that the function has the form

Even in our examples, we define the function in this way, but analytically it is possible to define the function implicitly, for example.

Try building this function yourself.

Did you manage?

Here's how I built it.

What equation did we end up with?

Right! Linear, which means that the graph will be a straight line. Let's make a table to determine which points belong to our line:

That's just what we were talking about ... One corresponds to several.

Let's try to draw what happened:

Is what we got a function?

That's right, no! Why? Try to answer this question with a picture. What did you get?

“Because one value corresponds to several values!”

What conclusion can we draw from this?

That's right, a function can't always be expressed explicitly, and what's "disguised" as a function isn't always a function!

Tabular way of defining a function

As the name suggests, this method is a simple plate. Yes Yes. Like the one we already made. For instance:

Here you immediately noticed a pattern - Y is three times larger than X. And now the “think very well” task: do you think that a function given in the form of a table is equivalent to a function?

Let's not talk for a long time, but let's draw!

So. We draw a function given in both ways:

Do you see the difference? It's not about the marked points! Take a closer look:

Have you seen it now? When we set the function in a tabular way, we reflect on the graph only those points that we have in the table and the line (as in our case) passes only through them. When we define a function in an analytical way, we can take any points, and our function is not limited to them. Here is such a feature. Remember!

Graphical way to build a function

The graphical way of constructing a function is no less convenient. We draw our function, and another interested person can find what y is equal to at a certain x, and so on. Graphical and analytical methods are among the most common.

However, here you need to remember what we talked about at the very beginning - not every “squiggle” drawn in the coordinate system is a function! Remembered? Just in case, I'll copy here the definition of what a function is:

As a rule, people usually name exactly those three ways of specifying a function that we have analyzed - analytical (using a formula), tabular and graphic, completely forgetting that a function can be described verbally. Like this? Yes, very easy!

Verbal description of the function

How to describe the function verbally? Let's take our recent example - . This function can be described as "each real value of x corresponds to its triple value." That's all. Nothing complicated. Of course, you will object - “there are such complex functions that it is simply impossible to set verbally!” Yes, there are some, but there are functions that are easier to describe verbally than to set with a formula. For example: "each natural value of x corresponds to the difference between the digits of which it consists, while the largest digit contained in the number entry is taken as the minuend." Now consider how our verbal description functions are implemented in practice:

The largest digit in a given number -, respectively, - is reduced, then:

Main types of functions

Now let's move on to the most interesting - consider the main types of functions with which you worked / work and will work in the course of school and institute mathematics, that is, we will get to know them, so to speak, and give them brief description. Read more about each function in the corresponding section.

Linear function

A function of the form, where, are real numbers.

The graph of this function is a straight line, so the construction of a linear function is reduced to finding the coordinates of two points.

The position of the straight line on the coordinate plane depends on the slope.

Function scope (aka argument range) - .

The range of values is .

quadratic function

Function of the form, where

The graph of the function is a parabola, when the branches of the parabola are directed downwards, when - upwards.

Many properties of a quadratic function depend on the value of the discriminant. The discriminant is calculated by the formula

The position of the parabola on the coordinate plane relative to the value and coefficient is shown in the figure:

Domain

The range of values depends on the extremum of the given function (the vertex of the parabola) and the coefficient (the direction of the branches of the parabola)

Inverse proportionality

The function given by the formula, where

The number is called the inverse proportionality factor. Depending on what value, the branches of the hyperbola are in different squares:

Domain - .

The range of values is .

SUMMARY AND BASIC FORMULA

1. A function is a rule according to which each element of a set is assigned a unique element of the set.

- this is a formula denoting a function, that is, the dependence of one variable on another;
- variable, or argument;
- dependent value - changes when the argument changes, that is, according to some specific formula that reflects the dependence of one value on another.

2. Valid argument values, or the scope of a function, is what is related to the possible under which the function makes sense.

3. Range of function values- this is what values it takes, with valid values.

4. There are 4 ways to set the function:

analytical (using formulas);
tabular;
graphic
verbal description.

5. Main types of functions:

: , where, are real numbers;
: , where;
: , where.

Schoolchildren are faced with the task of constructing a function graph at the very beginning of studying algebra and continue to build them from year to year. Starting from a graph of a linear function, for the construction of which you need to know only two points, to a parabola, for which you already need 6 points, a hyperbola and a sinusoid. Every year, the functions become more and more complex and it is no longer possible to plot their graphs according to a template, it is necessary to conduct more complex studies using derivatives and limits.

Let's figure out how to find the graph of a function? To do this, let's start with the simplest functions, the graphs of which are built by points, and then consider a plan for constructing more complex functions.

Plotting a Linear Function

To build the simplest graphs, a table of function values is used. The graph of a linear function is a straight line. Let's try to find the points of the graph of the function y=4x+5.

To do this, we take two arbitrary values of the variable x, substitute them one by one into the function, find the value of the variable y and put everything in the table.
Let's take the value x=0 and substitute it into the function instead of x - 0. We get: y=4*0+5, that is, y=5 write this value to the table under 0. Similarly, take x=0 we get y=4*1+5 , y=9.
Now, to build a function graph, you need to plot these points on the coordinate plane. Then you need to draw a straight line.

Plotting a Quadratic Function

A quadratic function is a function of the form y=ax 2 +bx +c, where x is a variable, a,b,c are numbers (a is not equal to 0). For example: y=x 2 , y=x 2 +5, y=(x-3) 2 , y=2x 2 +3x+5.

To build the simplest quadratic function y=x 2 usually take 5-7 points. Let's take the values for the variable x: -2, -1, 0, 1, 2 and find the values of y in the same way as when building the first graph.

The graph of a quadratic function is called a parabola. After constructing function graphs, students have new tasks associated with the graph.

Example 1: find the abscissa of the function graph point y=x 2 if the ordinate is 9. To solve the problem, you need to substitute its value 9 instead of y into the function. We get 9=x 2 and solve this equation. x=3 and x=-3. This can also be seen in the graph of the function.

Investigation of a function and construction of its graph

To plot more complex functions, you need to perform several steps aimed at its study. For this you need:

Find the scope of the function. The scope is all the values that x can take on. From the domain of definition, one should exclude those points at which the denominator becomes 0 or the radical expression becomes negative.
Set even or odd function. Recall that even is the function that meets the condition f(-x)=f(x). Its graph is symmetrical with respect to Oy. A function will be odd if it meets the condition f(-x)=-f(x). In this case, the graph is symmetrical about the origin.
Find points of intersection with coordinate axes. In order to find the abscissa of the point of intersection with the x-axis, it is necessary to solve the equation f(x)=0 (the ordinate is 0). To find the ordinate of the point of intersection with the Oy axis, it is necessary to substitute 0 in the function instead of the x variable (the abscissa is 0).
Find the asymptotes of the function. An asymptote is a line that the graph approaches indefinitely, but never crosses. Let's figure out how to find the asymptotes of the graph of a function.
- Vertical asymptote straight line of the form x=a
- Horizontal asymptote - a straight line of the form y \u003d a
- Oblique asymptote - straight line of the form y=kx+b
Find the extremum points of the function, the intervals of increase and decrease of the function. Find the extremum points of the function. To do this, you need to find the first derivative and equate it to 0. It is at these points that the function can change from increasing to decreasing. Let us determine the sign of the derivative on each interval. If the derivative is positive, then the graph of the function increases; if it is negative, it decreases.
Find the inflection points of the graph of the function, the intervals of convexity up and down.

Finding inflection points is now easier than ever. You just need to find the second derivative, then equate it to zero. Next, we find the sign of the second derivative on each interval. If positive, then the graph of the function is convex down, if negative - up.

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 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).

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 different kind 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.

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