What does the graph of a function depend on? Functions and graphs. Power function with non-integer rational or irrational exponent greater than one

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

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

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

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

Topics to review:

1. Graph the function

A familiar challenge! These met in OGE options mathematics. There they were considered difficult. But there is nothing complicated here.

Let's simplify the function formula:

Function graph - straight line with a punched out point

2. Graph the function

Let's select the integer part in the function formula:

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

Selection of the whole part - useful technique used in solving inequalities, plotting graphs, and evaluating integers in problems involving numbers and their properties. You will also meet him in the first year, when you have to take integrals.

3. Graph the function

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

4. Graph the function

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

We act in order:

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

2) squeeze 2 times horizontally,

3) stretch 3 times vertically,

4) move up by 1

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

5. Graph the function

Function scope:

Function zeros: and

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

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

Let's open the brackets in the function formula:

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

6. Graph the function

This is a fractional rational function.

Function scope

Function zeros: points - 3, 2, 6.

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

Vertical asymptotes:

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

Here is a sketch of the graph:

Another interesting technique is the addition of graphs.

7. Graph the function

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

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

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

8. Graph the function

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

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

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

9. Graph the function

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

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

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

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

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

10. Graph the function

The scope of the function is all real numbers, since

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

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

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

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

If or

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

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

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

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

The general scheme for constructing a graph of a function:

1. Function scope

2. Range of function values

3. Even - odd (if any)

4. Frequency (if any)

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

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

7. Asymptotes (if any).

8. Behavior of a function at infinity

9. Derivative of a function

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

A linear function is a function of the form y=kx+b, where x is an independent variable, k and b are any numbers.
schedule linear function is a straight line.

1. To plot a function graph, we need the coordinates of two points belonging to the graph of the function. To find them, you need to take two x values, substitute them into the equation of the function, and calculate the corresponding y values ​​from them.

For example, to plot the function y= x+2, it is convenient to take x=0 and x=3, then the ordinates of these points will be equal to y=2 and y=3. We get points A(0;2) and B(3;3). Let's connect them and get the graph of the function y= x+2:

2. In the formula y=kx+b, the number k is called the proportionality factor:
if k>0, then the function y=kx+b increases
if k
The coefficient b shows the shift of the graph of the function along the OY axis:
if b>0, then the graph of the function y=kx+b is obtained from the graph of the function y=kx by shifting b units up along the OY axis
if b
The figure below shows the graphs of the functions y=2x+3; y= ½x+3; y=x+3

Note that in all these functions the coefficient k Above zero, and functions are increasing. Moreover, the greater the value of k, the greater the angle of inclination of the straight line to the positive direction of the OX axis.

In all functions b=3 - and we see that all graphs intersect the OY axis at the point (0;3)

Now consider the graphs of functions y=-2x+3; y=- ½ x+3; y=-x+3

This time, in all functions, the coefficient k less than zero and features decrease. The coefficient b=3, and the graphs, as in the previous case, cross the OY axis at the point (0;3)

Consider the graphs of functions y=2x+3; y=2x; y=2x-3

Now, in all equations of functions, the coefficients k are equal to 2. And we got three parallel lines.

But the coefficients b are different, and these graphs intersect the OY axis at different points:
The graph of the function y=2x+3 (b=3) crosses the OY axis at the point (0;3)
The graph of the function y=2x (b=0) crosses the OY axis at the point (0;0) - the origin.
The graph of the function y=2x-3 (b=-3) crosses the OY axis at the point (0;-3)

So, if we know the signs of the coefficients k and b, then we can immediately imagine what the graph of the function y=kx+b looks like.
If a k 0

If a k>0 and b>0, then the graph of the function y=kx+b looks like:

If a k>0 and b, then the graph of the function y=kx+b looks like:

If a k, then the graph of the function y=kx+b looks like:

If a k=0, then the function y=kx+b turns into a function y=b and its graph looks like:

The ordinates of all points of the graph of the function y=b are equal to b If b=0, then the graph of the function y=kx (direct proportionality) passes through the origin:

3. Separately, we note the graph of the equation x=a. The graph of this equation is a straight line parallel to the OY axis, all points of which have an abscissa x=a.

For example, the graph of the equation x=3 looks like this:
Attention! The equation x=a is not a function, since one value of the argument corresponds to different values ​​of the function, which does not correspond to the definition of the function.

4. Condition for parallelism of two lines:

The graph of the function y=k 1 x+b 1 is parallel to the graph of the function y=k 2 x+b 2 if k 1 =k 2

5. The condition for two straight lines to be perpendicular:

The graph of the function y=k 1 x+b 1 is perpendicular to the graph of the function y=k 2 x+b 2 if k 1 *k 2 =-1 or k 1 =-1/k 2

6. Intersection points of the graph of the function y=kx+b with the coordinate axes.

with OY axis. The abscissa of any point belonging to the OY axis is equal to zero. Therefore, to find the point of intersection with the OY axis, you need to substitute zero instead of x in the equation of the function. We get y=b. That is, the point of intersection with the OY axis has coordinates (0;b).

With the x-axis: The ordinate of any point belonging to the x-axis is zero. Therefore, to find the point of intersection with the OX axis, you need to substitute zero instead of y in the equation of the function. We get 0=kx+b. Hence x=-b/k. That is, the point of intersection with the OX axis has coordinates (-b / k; 0):

The basic elementary functions, their inherent properties and the corresponding graphs are one of the basics of mathematical knowledge, similar in importance to the multiplication table. Elementary Functions are the basis, support for the study of all theoretical issues.

The article below provides key material on the topic of basic elementary functions. We will introduce terms, give them definitions; Let us study in detail each type of elementary functions and analyze their properties.

The following types of basic elementary functions are distinguished:

Definition 1

• constant function (constant);
• root of the nth degree;
• power function;
• exponential function;
• logarithmic function;
• trigonometric functions;
• fraternal trigonometric functions.

A constant function is defined by the formula: y = C (C is some real number) and also has a name: constant. This function determines whether any real value of the independent variable x corresponds to the same value of the variable y – the value C .

The graph of a constant is a straight line that is parallel to the x-axis and passes through a point having coordinates (0, C). For clarity, here are the graphs permanent functions y \u003d 5, y \u003d - 2, y \u003d 3, y \u003d 3 (indicated in black, red and blue in the drawing, respectively).

Definition 2

This elementary function is defined by the formula y = x n (n - natural number more than one).

Let's consider two variations of the function.

1. Root of the nth degree, n is an even number

For clarity, we indicate the drawing, which shows the graphs of such functions: y = x , y = x 4 and y = x 8 . These functions are color-coded: black, red and blue, respectively.

A similar view of the graphs of the function of an even degree for other values ​​​​of the indicator.

Definition 3

Properties of the function root of the nth degree, n is an even number

• the domain of definition is the set of all non-negative real numbers [ 0 , + ∞) ;
• when x = 0 , the function y = x n has a value equal to zero;
• given function - function general form (is neither even nor odd);
• range: [ 0 , + ∞) ;
• this function y = x n with even exponents of the root increases over the entire domain of definition;
• the function has a convexity with an upward direction over the entire domain of definition;
• there are no inflection points;
• there are no asymptotes;
• the graph of the function for even n passes through the points (0 ; 0) and (1 ; 1) .

1. Root of the nth degree, n is an odd number

Such a function is defined on the entire set of real numbers. For clarity, consider the graphs of functions y = x 3 , y = x 5 and x 9 . In the drawing, they are indicated by colors: black, red and blue colors of the curves, respectively.

Other odd values ​​of the exponent of the root of the function y = x n will give a graph of a similar form.

Definition 4

Properties of the function root of the nth degree, n is an odd number

• the domain of definition is the set of all real numbers;
• this function is odd;
• the range of values ​​is the set of all real numbers;
• the function y = x n with odd exponents of the root increases over the entire domain of definition;
• the function has concavity on the interval (- ∞ ; 0 ] and convexity on the interval [ 0 , + ∞) ;
• the inflection point has coordinates (0 ; 0) ;
• there are no asymptotes;
• the graph of the function for odd n passes through the points (- 1 ; - 1) , (0 ; 0) and (1 ; 1) .

Power function

Definition 5

The power function is defined by the formula y = x a .

The type of graphs and properties of the function depend on the value of the exponent.

• when a power function has an integer exponent a, then the form of the graph of the power function and its properties depend on whether the exponent is even or odd, and also what sign the exponent has. Let us consider all these special cases in more detail below;
• the exponent can be fractional or irrational - depending on this, the type of graphs and the properties of the function also vary. We will analyze special cases by setting several conditions: 0< a < 1 ; a > 1 ; - 1 < a < 0 и a < - 1 ;
• a power function can have a zero exponent, we will also analyze this case in more detail below.

Let's analyze the power function y = x a when a is an odd positive number, for example, a = 1 , 3 , 5 …

For clarity, we indicate the graphs of such power functions: y = x (black color of the graph), y = x 3 (blue color of the chart), y = x 5 (red color of the graph), y = x 7 (green graph). When a = 1 , we get a linear function y = x .

Definition 6

Properties of a power function when the exponent is an odd positive

• the function is increasing for x ∈ (- ∞ ; + ∞) ;
• the function is convex for x ∈ (- ∞ ; 0 ] and concave for x ∈ [ 0 ; + ∞) (excluding the linear function);
• the inflection point has coordinates (0 ; 0) (excluding the linear function);
• there are no asymptotes;
• function passing points: (- 1 ; - 1) , (0 ; 0) , (1 ; 1) .

Let's analyze the power function y = x a when a is an even positive number, for example, a = 2 , 4 , 6 ...

For clarity, we indicate the graphs of such power functions: y \u003d x 2 (black color of the graph), y = x 4 (blue color of the graph), y = x 8 (red color of the graph). When a = 2, we get a quadratic function whose graph is a quadratic parabola.

Definition 7

Properties of a power function when the exponent is even positive:

• domain of definition: x ∈ (- ∞ ; + ∞) ;
• decreasing for x ∈ (- ∞ ; 0 ] ;
• the function is concave for x ∈ (- ∞ ; + ∞) ;
• there are no inflection points;
• there are no asymptotes;
• function passing points: (- 1 ; 1) , (0 ; 0) , (1 ; 1) .

The figure below shows examples of exponential function graphs y = x a when a is an odd negative number: y = x - 9 (black color of the graph); y = x - 5 (blue color of the graph); y = x - 3 (red color of the chart); y = x - 1 (green graph). When a \u003d - 1, we get an inverse proportionality, the graph of which is a hyperbola.

Definition 8

Power function properties when the exponent is odd negative:

When x \u003d 0, we get a discontinuity of the second kind, since lim x → 0 - 0 x a \u003d - ∞, lim x → 0 + 0 x a \u003d + ∞ for a \u003d - 1, - 3, - 5, .... Thus, the straight line x = 0 is a vertical asymptote;

• range: y ∈ (- ∞ ; 0) ∪ (0 ; + ∞) ;
• the function is odd because y (- x) = - y (x) ;
• the function is decreasing for x ∈ - ∞ ; 0 ∪ (0 ; + ∞) ;
• the function is convex for x ∈ (- ∞ ; 0) and concave for x ∈ (0 ; + ∞) ;
• there are no inflection points;

k = lim x → ∞ x a x = 0 , b = lim x → ∞ (x a - k x) = 0 ⇒ y = k x + b = 0 when a = - 1 , - 3 , - 5 , . . . .

• function passing points: (- 1 ; - 1) , (1 ; 1) .

The figure below shows examples of power function graphs y = x a when a is an even negative number: y = x - 8 (chart in black); y = x - 4 (blue color of the graph); y = x - 2 (red color of the graph).

Definition 9

Power function properties when the exponent is even negative:

• domain of definition: x ∈ (- ∞ ; 0) ∪ (0 ; + ∞) ;

When x \u003d 0, we get a discontinuity of the second kind, since lim x → 0 - 0 x a \u003d + ∞, lim x → 0 + 0 x a \u003d + ∞ for a \u003d - 2, - 4, - 6, .... Thus, the straight line x = 0 is a vertical asymptote;

• the function is even because y (- x) = y (x) ;
• the function is increasing for x ∈ (- ∞ ; 0) and decreasing for x ∈ 0 ; +∞ ;
• the function is concave for x ∈ (- ∞ ; 0) ∪ (0 ; + ∞) ;
• there are no inflection points;
• the horizontal asymptote is a straight line y = 0 because:

k = lim x → ∞ x a x = 0 , b = lim x → ∞ (x a - k x) = 0 ⇒ y = k x + b = 0 when a = - 2 , - 4 , - 6 , . . . .

• function passing points: (- 1 ; 1) , (1 ; 1) .

From the very beginning, pay attention to the following aspect: in the case when a is a positive fraction with an odd denominator, some authors take the interval - ∞ as the domain of definition of this power function; + ∞ , stipulating that the exponent a is an irreducible fraction. On the this moment the authors of many textbooks on algebra and the beginnings of analysis DO NOT DEFINE power functions, where the exponent is a fraction with an odd denominator for negative values ​​of the argument. Further, we will adhere to just such a position: we take the set [ 0 ; +∞) . Recommendation for students: find out the teacher's point of view at this point in order to avoid disagreements.

So let's take a look at the power function y = x a when the exponent is a rational or irrational number provided that 0< a < 1 .

Let us illustrate with graphs the power functions y = x a when a = 11 12 (chart in black); a = 5 7 (red color of the graph); a = 1 3 (blue color of the chart); a = 2 5 (green color of the graph).

Other values ​​of the exponent a (assuming 0< a < 1) дадут аналогичный вид графика.

Definition 10

Power function properties at 0< a < 1:

• range: y ∈ [ 0 ; +∞) ;
• the function is increasing for x ∈ [ 0 ; +∞) ;
• the function has convexity for x ∈ (0 ; + ∞) ;
• there are no inflection points;
• there are no asymptotes;

Let's analyze the power function y = x a when the exponent is a non-integer rational or irrational number provided that a > 1 .

We illustrate the graphs of the power function y \u003d x a under given conditions using the following functions as an example: y \u003d x 5 4, y \u003d x 4 3, y \u003d x 7 3, y \u003d x 3 π (black, red, blue, green graphs, respectively).

Other values ​​of the exponent a under the condition a > 1 will give a similar view of the graph.

Definition 11

Power function properties for a > 1:

• domain of definition: x ∈ [ 0 ; +∞) ;
• range: y ∈ [ 0 ; +∞) ;
• this function is a function of general form (it is neither odd nor even);
• the function is increasing for x ∈ [ 0 ; +∞) ;
• the function is concave for x ∈ (0 ; + ∞) (when 1< a < 2) и выпуклость при x ∈ [ 0 ; + ∞) (когда a > 2);
• there are no inflection points;
• there are no asymptotes;
• function passing points: (0 ; 0) , (1 ; 1) .

We draw your attention! When a is a negative fraction with an odd denominator, in the works of some authors there is a view that the domain of definition in this case– interval - ∞ ; 0 ∪ (0 ; + ∞) with the proviso that the exponent a is an irreducible fraction. At the moment the authors teaching materials according to algebra and the beginnings of analysis, power functions with an exponent in the form of a fraction with an odd denominator with negative values ​​of the argument are NOT DEFINED. Further, we adhere to just such a view: we take the set (0 ; + ∞) as the domain of power functions with fractional negative exponents. Suggestion for students: Clarify your teacher's vision at this point to avoid disagreement.

We continue the topic and analyze the power function y = x a provided: - 1< a < 0 .

Here is a drawing of graphs of the following functions: y = x - 5 6 , y = x - 2 3 , y = x - 1 2 2 , y = x - 1 7 (black, red, blue, green lines, respectively).

Definition 12

Power function properties at - 1< a < 0:

lim x → 0 + 0 x a = + ∞ when - 1< a < 0 , т.е. х = 0 – вертикальная асимптота;

• range: y ∈ 0 ; +∞ ;
• this function is a function of general form (it is neither odd nor even);
• there are no inflection points;

The drawing below shows graphs of power functions y = x - 5 4 , y = x - 5 3 , y = x - 6 , y = x - 24 7 (black, red, blue, green colors curves, respectively).

Definition 13

Power function properties for a< - 1:

• domain of definition: x ∈ 0 ; +∞ ;

lim x → 0 + 0 x a = + ∞ when a< - 1 , т.е. х = 0 – вертикальная асимптота;

• range: y ∈ (0 ; + ∞) ;
• this function is a function of general form (it is neither odd nor even);
• the function is decreasing for x ∈ 0; +∞ ;
• the function is concave for x ∈ 0; +∞ ;
• there are no inflection points;
• horizontal asymptote - straight line y = 0 ;
• function passing point: (1 ; 1) .

When a \u003d 0 and x ≠ 0, we get the function y \u003d x 0 \u003d 1, which determines the line from which the point (0; 1) is excluded (we agreed that the expression 0 0 will not be given any value).

The exponential function has the form y = a x , where a > 0 and a ≠ 1 , and the graph of this function looks different based on the value of the base a . Let's consider special cases.

First, let's analyze the situation when the base of the exponential function has a value from zero to one (0< a < 1) . An illustrative example is the graphs of functions for a = 1 2 (blue color of the curve) and a = 5 6 (red color of the curve).

The graphs of the exponential function will have a similar form for other values ​​​​of the base, provided that 0< a < 1 .

Definition 14

Properties of an exponential function when the base is less than one:

• range: y ∈ (0 ; + ∞) ;
• this function is a function of general form (it is neither odd nor even);
• an exponential function whose base is less than one is decreasing over the entire domain of definition;
• there are no inflection points;
• the horizontal asymptote is the straight line y = 0 with the variable x tending to + ∞ ;

Now consider the case when the base of the exponential function is greater than one (a > 1).

Let's illustrate this special case with the graph of exponential functions y = 3 2 x (blue color of the curve) and y = e x (red color of the graph).

Other values ​​​​of the base, greater than one, will give a similar view of the graph of the exponential function.

Definition 15

Properties of the exponential function when the base is greater than one:

• the domain of definition is the entire set of real numbers;
• range: y ∈ (0 ; + ∞) ;
• this function is a function of general form (it is neither odd nor even);
• an exponential function whose base is greater than one is increasing for x ∈ - ∞ ; +∞ ;
• the function is concave for x ∈ - ∞ ; +∞ ;
• there are no inflection points;
• horizontal asymptote - straight line y = 0 with variable x tending to - ∞ ;
• function passing point: (0 ; 1) .

The logarithmic function has the form y = log a (x) , where a > 0 , a ≠ 1 .

Such a function is defined only for positive values ​​of the argument: for x ∈ 0 ; +∞ .

The plot of the logarithmic function has different kind, based on the value of the base a.

Consider first the situation when 0< a < 1 . Продемонстрируем этот частный случай графиком логарифмической функции при a = 1 2 (синий цвет кривой) и а = 5 6 (красный цвет кривой).

Other values ​​of the base, not greater than one, will give a similar view of the graph.

Definition 16

Properties of a logarithmic function when the base is less than one:

• domain of definition: x ∈ 0 ; +∞ . As x tends to zero from the right, the values ​​of the function tend to + ∞;
• range: y ∈ - ∞ ; +∞ ;
• this function is a function of general form (it is neither odd nor even);
• logarithmic
• the function is concave for x ∈ 0; +∞ ;
• there are no inflection points;
• there are no asymptotes;

Now let's analyze a special case when the base of the logarithmic function is greater than one: a > 1 . In the drawing below, there are graphs of logarithmic functions y = log 3 2 x and y = ln x (blue and red colors of the graphs, respectively).

Other values ​​of the base greater than one will give a similar view of the graph.

Definition 17

Properties of a logarithmic function when the base is greater than one:

• domain of definition: x ∈ 0 ; +∞ . As x tends to zero from the right, the values ​​of the function tend to - ∞;
• range: y ∈ - ∞ ; + ∞ (the whole set of real numbers);
• this function is a function of general form (it is neither odd nor even);
• the logarithmic function is increasing for x ∈ 0; +∞ ;
• the function has convexity for x ∈ 0; +∞ ;
• there are no inflection points;
• there are no asymptotes;
• function passing point: (1 ; 0) .

Trigonometric functions are sine, cosine, tangent and cotangent. Let's analyze the properties of each of them and the corresponding graphs.

In general, all trigonometric functions are characterized by the property of periodicity, i.e. when the function values ​​are repeated at different meanings argument, differing from each other by the value of the period f (x + T) = f (x) (T is the period). Thus, the item "least positive period" is added to the list of properties of trigonometric functions. In addition, we will indicate such values ​​of the argument for which the corresponding function vanishes.

1. Sine function: y = sin(x)

The graph of this function is called a sine wave.

Definition 18

Properties of the sine function:

• domain of definition: the whole set of real numbers x ∈ - ∞ ; +∞ ;
• the function vanishes when x = π k , where k ∈ Z (Z is the set of integers);
• the function is increasing for x ∈ - π 2 + 2 π · k ; π 2 + 2 π k , k ∈ Z and decreasing for x ∈ π 2 + 2 π k ; 3 π 2 + 2 π k , k ∈ Z ;
• the sine function has local maxima at the points π 2 + 2 π · k ; 1 and local minima at points - π 2 + 2 π · k ; - 1 , k ∈ Z ;
• the sine function is concave when x ∈ - π + 2 π k; 2 π k , k ∈ Z and convex when x ∈ 2 π k ; π + 2 π k , k ∈ Z ;
• there are no asymptotes.

1. cosine function: y=cos(x)

The graph of this function is called a cosine wave.

Definition 19

Properties of the cosine function:

• domain of definition: x ∈ - ∞ ; +∞ ;
• the smallest positive period: T \u003d 2 π;
• range: y ∈ - 1 ; one ;
• this function is even, since y (- x) = y (x) ;
• the function is increasing for x ∈ - π + 2 π · k ; 2 π · k , k ∈ Z and decreasing for x ∈ 2 π · k ; π + 2 π k , k ∈ Z ;
• the cosine function has local maxima at points 2 π · k ; 1 , k ∈ Z and local minima at the points π + 2 π · k ; - 1 , k ∈ z ;
• the cosine function is concave when x ∈ π 2 + 2 π · k ; 3 π 2 + 2 π k , k ∈ Z and convex when x ∈ - π 2 + 2 π k ; π 2 + 2 π · k , k ∈ Z ;
• inflection points have coordinates π 2 + π · k ; 0 , k ∈ Z
• there are no asymptotes.

1. Tangent function: y = t g (x)

The graph of this function is called tangentoid.

Definition 20

Properties of the tangent function:

• domain of definition: x ∈ - π 2 + π · k ; π 2 + π k , where k ∈ Z (Z is the set of integers);
• The behavior of the tangent function on the boundary of the domain of definition lim x → π 2 + π · k + 0 t g (x) = - ∞ , lim x → π 2 + π · k - 0 t g (x) = + ∞ . Thus, the lines x = π 2 + π · k k ∈ Z are vertical asymptotes;
• the function vanishes when x = π k for k ∈ Z (Z is the set of integers);
• range: y ∈ - ∞ ; +∞ ;
• this function is odd because y (- x) = - y (x) ;
• the function is increasing at - π 2 + π · k ; π 2 + π k , k ∈ Z ;
• the tangent function is concave for x ∈ [ π · k ; π 2 + π k) , k ∈ Z and convex for x ∈ (- π 2 + π k ; π k ] , k ∈ Z ;
• inflection points have coordinates π k; 0 , k ∈ Z ;

1. Cotangent function: y = c t g (x)

The graph of this function is called the cotangentoid. .

Definition 21

Properties of the cotangent function:

• domain of definition: x ∈ (π k ; π + π k) , where k ∈ Z (Z is the set of integers);

Behavior of the cotangent function on the boundary of the domain of definition lim x → π · k + 0 t g (x) = + ∞ , lim x → π · k - 0 t g (x) = - ∞ . Thus, the lines x = π k k ∈ Z are vertical asymptotes;

• the smallest positive period: T \u003d π;
• the function vanishes when x = π 2 + π k for k ∈ Z (Z is the set of integers);
• range: y ∈ - ∞ ; +∞ ;
• this function is odd because y (- x) = - y (x) ;
• the function is decreasing for x ∈ π · k ; π + π k , k ∈ Z ;
• the cotangent function is concave for x ∈ (π k ; π 2 + π k ] , k ∈ Z and convex for x ∈ [ - π 2 + π k ; π k) , k ∈ Z ;
• inflection points have coordinates π 2 + π · k ; 0 , k ∈ Z ;
• there are no oblique and horizontal asymptotes.

The inverse trigonometric functions are the arcsine, arccosine, arctangent, and arccotangent. Often, due to the presence of the prefix "arc" in the name, inverse trigonometric functions are called arc functions. .

1. Arcsine function: y = a r c sin (x)

Definition 22

Properties of the arcsine function:

• this function is odd because y (- x) = - y (x) ;
• the arcsine function is concave for x ∈ 0; 1 and convexity for x ∈ - 1 ; 0;
• inflection points have coordinates (0 ; 0) , it is also the zero of the function;
• there are no asymptotes.

1. Arccosine function: y = a r c cos (x)

Definition 23

Arccosine function properties:

• domain of definition: x ∈ - 1 ; one ;
• range: y ∈ 0 ; π;
• this function is of general form (neither even nor odd);
• the function is decreasing on the entire domain of definition;
• the arccosine function is concave for x ∈ - 1 ; 0 and convexity for x ∈ 0 ; one ;
• inflection points have coordinates 0 ; π2;
• there are no asymptotes.

1. Arctangent function: y = a r c t g (x)

Definition 24

Arctangent function properties:

• domain of definition: x ∈ - ∞ ; +∞ ;
• range: y ∈ - π 2 ; π2;
• this function is odd because y (- x) = - y (x) ;
• the function is increasing over the entire domain of definition;
• the arctangent function is concave for x ∈ (- ∞ ; 0 ] and convex for x ∈ [ 0 ; + ∞) ;
• the inflection point has coordinates (0; 0), it is also the zero of the function;
• horizontal asymptotes are straight lines y = - π 2 for x → - ∞ and y = π 2 for x → + ∞ (the asymptotes in the figure are green lines).

1. Arc cotangent function: y = a r c c t g (x)

Definition 25

Arc cotangent function properties:

• domain of definition: x ∈ - ∞ ; +∞ ;
• range: y ∈ (0 ; π) ;
• this function is of a general type;
• the function is decreasing on the entire domain of definition;
• the arc cotangent function is concave for x ∈ [ 0 ; + ∞) and convexity for x ∈ (- ∞ ; 0 ] ;
• the inflection point has coordinates 0 ; π2;
• horizontal asymptotes are straight lines y = π at x → - ∞ (green line in the drawing) and y = 0 at x → + ∞.

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

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:

1. , since the root expression must be greater than or equal to zero.
2. , since it is impossible to divide by zero and the radical expression cannot be negative.
3. , since, respectively, for all.
4. 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 a very, very important element of our definition. I will try to explain to you on the fingers.

Let's say we have a function given by a straight line. . When, we substitute this value into our "rule" and get that. One value corresponds to one value. We can even make a table different meanings and build a graph of this function to make sure of 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 real 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 AT) 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:

1. , if
2. , 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?

Correctly! 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 example:

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 figure in given number- , respectively, - reduced, then:

Main types of functions

Now let's move on to the most interesting - we will 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 a 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.

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 xО [-3;3] , on the interval xО [-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) ОR, if n is an odd number and D(x)= , on the interval xО , on the interval xО [-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 xн ; 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.