1. Linear fractional function and its graph

A function of the form y = P(x) / Q(x), where P(x) and Q(x) are polynomials, is called a fractional rational function.

You are probably already familiar with the concept of rational numbers. Similarly rational functions are functions that can be represented as a quotient of two polynomials.

If a fractional rational function is a quotient of two linear functions - polynomials of the first degree, i.e. view function

y = (ax + b) / (cx + d), then it is called fractional linear.

Note that in the function y = (ax + b) / (cx + d), c ≠ 0 (otherwise the function becomes linear y = ax/d + b/d) and that a/c ≠ b/d (otherwise the function is a constant ). The linear-fractional function is defined for all real numbers, except for x = -d/c. Graphs of linear-fractional functions do not differ in form from the graph you know y = 1/x. The curve that is the graph of the function y = 1/x is called hyperbole. With an unlimited increase in x in absolute value, the function y = 1/x decreases indefinitely in absolute value and both branches of the graph approach the abscissa axis: the right one approaches from above, and the left one approaches from below. The lines approached by the branches of a hyperbola are called its asymptotes.

Example 1

y = (2x + 1) / (x - 3).

Decision.

Let's select the integer part: (2x + 1) / (x - 3) = 2 + 7 / (x - 3).

Now it is easy to see that the graph of this function is obtained from the graph of the function y = 1/x by the following transformations: shift by 3 unit segments to the right, stretch along the Oy axis by 7 times and shift by 2 unit segments up.

Any fraction y = (ax + b) / (cx + d) can be written in the same way, highlighting the “whole part”. Consequently, the graphs of all linear-fractional functions are hyperbolas shifted along the coordinate axes in various ways and stretched along the Oy axis.

To plot a graph of some arbitrary linear-fractional function, it is not at all necessary to transform the fraction that defines this function. Since we know that the graph is a hyperbola, it will be enough to find the lines to which its branches approach - the hyperbola asymptotes x = -d/c and y = a/c.

Example 2

Find the asymptotes of the graph of the function y = (3x + 5)/(2x + 2).

Decision.

The function is not defined, when x = -1. Hence, the line x = -1 serves as a vertical asymptote. To find the horizontal asymptote, let's find out what the values ​​of the function y(x) approach when the argument x increases in absolute value.

To do this, we divide the numerator and denominator of the fraction by x:

y = (3 + 5/x) / (2 + 2/x).

As x → ∞ the fraction tends to 3/2. Hence, the horizontal asymptote is the straight line y = 3/2.

Example 3

Plot the function y = (2x + 1)/(x + 1).

Decision.

We select the “whole part” of the fraction:

(2x + 1) / (x + 1) = (2x + 2 - 1) / (x + 1) = 2(x + 1) / (x + 1) - 1/(x + 1) =

2 – 1/(x + 1).

Now it is easy to see that the graph of this function is obtained from the graph of the function y = 1/x by the following transformations: a shift of 1 unit to the left, a symmetric display with respect to Ox, and a shift of 2 unit intervals up along the Oy axis.

Domain of definition D(y) = (-∞; -1)ᴗ(-1; +∞).

Range of values ​​E(y) = (-∞; 2)ᴗ(2; +∞).

Intersection points with axes: c Oy: (0; 1); c Ox: (-1/2; 0). The function increases on each of the intervals of the domain of definition.

Answer: figure 1.

2. Fractional-rational function

Consider a fractional rational function of the form y = P(x) / Q(x), where P(x) and Q(x) are polynomials of degree higher than the first.

Examples of such rational functions:

y \u003d (x 3 - 5x + 6) / (x 7 - 6) or y \u003d (x - 2) 2 (x + 1) / (x 2 + 3).

If the function y = P(x) / Q(x) is a quotient of two polynomials of degree higher than the first, then its graph will, as a rule, be more complicated, and it can sometimes be difficult to build it exactly, with all the details. However, it is often enough to apply techniques similar to those with which we have already met above.

Let the fraction be proper (n< m). Известно, что любую несократимую рациональную дробь можно представить, и притом единственным образом, в виде суммы конечного числа элементарных дробей, вид которых определяется разложением знаменателя дроби Q(x) в произведение действительных сомножителей:

P(x) / Q(x) \u003d A 1 / (x - K 1) m1 + A 2 / (x - K 1) m1-1 + ... + A m1 / (x - K 1) + ... +

L 1 /(x – K s) ms + L 2 /(x – K s) ms-1 + … + L ms /(x – K s) + …+

+ (B 1 x + C 1) / (x 2 +p 1 x + q 1) m1 + … + (B m1 x + C m1) / (x 2 +p 1 x + q 1) + …+

+ (M 1 x + N 1) / (x 2 + p t x + q t) m1 + ... + (M m1 x + N m1) / (x 2 + p t x + q t).

Obviously, the graph of a fractional rational function can be obtained as the sum of graphs of elementary fractions.

Plotting fractional rational functions

Consider several ways to plot a fractional-rational function.

Example 4

Plot the function y = 1/x 2 .

Decision.

We use the graph of the function y \u003d x 2 to plot the graph y \u003d 1 / x 2 and use the method of "dividing" the graphs.

Domain D(y) = (-∞; 0)ᴗ(0; +∞).

Range of values ​​E(y) = (0; +∞).

There are no points of intersection with the axes. The function is even. Increases for all x from the interval (-∞; 0), decreases for x from 0 to +∞.

Answer: figure 2.

Example 5

Plot the function y = (x 2 - 4x + 3) / (9 - 3x).

Decision.

Domain D(y) = (-∞; 3)ᴗ(3; +∞).

y \u003d (x 2 - 4x + 3) / (9 - 3x) \u003d (x - 3) (x - 1) / (-3 (x - 3)) \u003d - (x - 1) / 3 \u003d -x / 3 + 1/3.

Here we used the technique of factoring, reduction and reduction to a linear function.

Answer: figure 3.

Example 6

Plot the function y \u003d (x 2 - 1) / (x 2 + 1).

Decision.

The domain of definition is D(y) = R. Since the function is even, the graph is symmetrical about the y-axis. Before plotting, we again transform the expression by highlighting the integer part:

y \u003d (x 2 - 1) / (x 2 + 1) \u003d 1 - 2 / (x 2 + 1).

Note that the selection of the integer part in the formula of a fractional-rational function is one of the main ones when plotting graphs.

If x → ±∞, then y → 1, i.e., the line y = 1 is a horizontal asymptote.

Answer: figure 4.

Example 7

Consider the function y = x/(x 2 + 1) and try to find exactly its largest value, i.e. the highest point on the right half of the graph. To accurately build this graph, today's knowledge is not enough. It is obvious that our curve cannot "climb" very high, since the denominator quickly begins to “overtake” the numerator. Let's see if the value of the function can be equal to 1. To do this, you need to solve the equation x 2 + 1 \u003d x, x 2 - x + 1 \u003d 0. This equation has no real roots. So our assumption is wrong. To find the largest value of the function, you need to find out for which largest A the equation A \u003d x / (x 2 + 1) will have a solution. Let's replace the original equation with a quadratic one: Ax 2 - x + A \u003d 0. This equation has a solution when 1 - 4A 2 ≥ 0. From here we find the largest value A \u003d 1/2.

Answer: Figure 5, max y(x) = ½.

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Build a function

We bring to your attention a service for plotting function graphs online, all rights to which belong to the company Desmos. Use the left column to enter functions. You can enter manually or using the virtual keyboard at the bottom of the window. To enlarge the chart window, you can hide both the left column and the virtual keyboard.

Benefits of online charting

• Visual display of introduced functions
• Building very complex graphs
• Plotting implicitly defined graphs (e.g. ellipse x^2/9+y^2/16=1)
• The ability to save charts and get a link to them, which becomes available to everyone on the Internet
• Scale control, line color
• The ability to plot graphs by points, the use of constants
• Construction of several graphs of functions at the same time
• Plotting in polar coordinates (use r and θ(\theta))

With us it is easy to build graphs of varying complexity online. The construction is done instantly. The service is in demand for finding intersection points of functions, for displaying graphs for their further transfer to a Word document as illustrations for solving problems, for analyzing the behavioral features of function graphs. The best browser for working with charts on this page of the site is Google Chrome. When using other browsers, correct operation is not guaranteed.

Once you really understand what a function is (you may have to read the lesson more than once), you will be able to solve problems with functions with more confidence.

In this lesson, we will analyze how to solve the main types of function problems and function graphs.

How to get the value of a function

Let's consider the task. The function is given by the formula " y \u003d 2x - 1"

1. Calculate " y"When" x \u003d 15"
2. Find the value " x", At which the value " y "is equal to" −19 ".

In order to calculate " y"With" x \u003d 15"It is enough to substitute the required numerical value into the function instead of" x".

The solution entry looks like this:

y(15) = 2 15 - 1 = 30 - 1 = 29

In order to find " x"According to the known" y", It is necessary to substitute a numerical value instead of" y "in the function formula.

That is, now, on the contrary, to search for " x"We substitute in the function" y \u003d 2x - 1 "Instead of" y ", the number" −19".

−19 = 2x − 1

We have obtained a linear equation with an unknown "x", which is solved according to the rules for solving linear equations.

Remember!

Don't forget about the transfer rule in equations.

When transferring from the left side of the equation to the right (and vice versa), the letter or number changes sign to opposite.

−19 = 2x − 1
0 = 2x − 1 + 19
-2x = -1 + 19
−2x = 18

As with solving a linear equation, to find the unknown, now we need to multiply both left and right side to "−1" to change the sign.

-2x = 18 | (−1)
2x = −18

Now let's divide both the left and right sides by "2" to find "x".

2x = 18 | (:2)
x=9

How to check if equality is true for a function

Let's consider the task. The function is given by the formula "f(x) = 2 − 5x".

Is the equality "f(−2) = −18" true?

To check whether the equality is true, you need to substitute the numerical value “x = −2" into the function " f (x) \u003d 2 - 5x"And compare with what happens in the calculations.

Important!

When you substitute a negative number for "x", be sure to enclose it in brackets.

Not right

Correctly

With the help of calculations, we got "f(−2) = 12".

This means that "f(−2) = −18" for the function "f(x) = 2 − 5x" is not a valid equality.

How to check if a point belongs to a graph of a function

Consider the function " y \u003d x 2 −5x + 6"

It is required to find out whether the point with coordinates (1; 2) belongs to the graph of this function.

For this task, there is no need to plot a given function.

Remember!

To determine whether a point belongs to a function, it is enough to substitute its coordinates into the function (coordinate along the axis " Ox" Instead of " x"And the coordinate along the axis" Oy "Instead of" y ").

If this works out true equality, so the point belongs to the function.

Let's return to our task. Substitute in the function "y \u003d x 2 - 5x + 6" the coordinates of the point (1; 2).

Instead of " x"We substitute" 1". Instead of " y"Substitute" 2».

2 = 1 2 − 5 1 + 6
2 = 1 − 5 + 6
2 = −4 + 6
2 = 2 (correct)

We have obtained the correct equality, which means that the point with coordinates (1; 2) belongs to the given function.

Now let's check the point with coordinates (0; 1) . Does she belong
functions "y \u003d x 2 - 5x + 6"?

Instead of "x", let's substitute "0". Instead of " y"Substitute" 1».

1 = 0 2 − 5 0 + 6
1 = 0 − 0 + 6
1 = 6 (wrong)

In this case, we did not get the correct equality. This means that the point with coordinates (0; 1) does not belong to the function " y \u003d x 2 - 5x + 6 "

How to get function point coordinates

From any function graph, you can take the coordinates of a point. Then you need to make sure that when substituting the coordinates in the function formula, the correct equality is obtained.

Consider the function "y(x) = −2x + 1". We have already built its schedule in the previous lesson.

Let's find on the graph of the function " y (x) \u003d -2x + 1", which is equal to" y"For x \u003d 2.

To do this, from the value " 2"On the axis" Ox", Draw a perpendicular to the graph of the function. From the point of intersection of the perpendicular and the graph of the function, draw another perpendicular to the axis "Oy".

The resulting value " −3"On the axis" Oy"And will be the desired value" y».

Let's make sure that we correctly took the coordinates of the point for x = 2
in the function "y(x) = −2x + 1".

To do this, we substitute x \u003d 2 into the formula of the function "y (x) \u003d -2x + 1". If we draw the perpendicular correctly, we should also end up with y = −3 .

y(2) = -2 2 + 1 = -4 + 1 = -3

When calculating, we also got y = −3.

This means that we correctly received the coordinates from the graph of the function.

Important!

Be sure to check all the coordinates of the point from the function graph by substituting the values ​​\u200b\u200bof "x" into the function.

When substituting the numeric value "x" into the function, the result should be the same value" y", which you got on the chart.

When obtaining the coordinates of points from the graph of the function, it is highly likely that you will make a mistake, because drawing a perpendicular to the axes is performed "by eye".

Only substituting values ​​into a function formula gives accurate results.

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.

Page navigation.

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).

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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.

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.

Let's see how to explore a function using a graph. It turns out that looking at the graph, you can find out everything that interests us, namely:

• function scope
• function range
• function zeros
• periods of increase and decrease
• high and low points
• the largest and smallest value of the function on the segment.

Let's clarify the terminology:

Abscissa is the horizontal coordinate of the point.
Ordinate- vertical coordinate.
abscissa- the horizontal axis, most often called the axis.
Y-axis- vertical axis, or axis.

Argument is an independent variable on which the values ​​of the function depend. Most often indicated.
In other words, we ourselves choose , substitute in the function formula and get .

Domain functions - the set of those (and only those) values ​​of the argument for which the function exists.
Denoted: or .

In our figure, the domain of the function is a segment. It is on this segment that the graph of the function is drawn. Only here this function exists.

Function range is the set of values ​​that the variable takes. In our figure, this is a segment - from the lowest to the highest value.

Function zeros- points where the value of the function is equal to zero, i.e. . In our figure, these are the points and .

Function values ​​are positive where . In our figure, these are the intervals and .
Function values ​​are negative where . We have this interval (or interval) from to.

The most important concepts - increasing and decreasing functions on some set. As a set, you can take a segment, an interval, a union of intervals, or the entire number line.

Function increases

In other words, the more , the more , that is, the graph goes to the right and up.

Function decreases on the set if for any and belonging to the set the inequality implies the inequality .

For a decreasing function, a larger value corresponds to a smaller value. The graph goes right and down.

In our figure, the function increases on the interval and decreases on the intervals and .

Let's define what is maximum and minimum points of the function.

Maximum point- this is an internal point of the domain of definition, such that the value of the function in it is greater than in all points sufficiently close to it.
In other words, the maximum point is such a point, the value of the function at which more than in neighboring ones. This is a local "hill" on the chart.

In our figure - the maximum point.

Low point- an internal point of the domain of definition, such that the value of the function in it is less than in all points sufficiently close to it.
That is, the minimum point is such that the value of the function in it is less than in neighboring ones. On the graph, this is a local “hole”.

In our figure - the minimum point.

The point is the boundary. It is not an interior point of the domain of definition and therefore does not fit the definition of a maximum point. After all, she has no neighbors on the left. In the same way, there can be no minimum point on our chart.

The maximum and minimum points are collectively called extremum points of the function. In our case, this is and .

But what if you need to find, for example, function minimum on the cut? In this case, the answer is: because function minimum is its value at the minimum point.

Similarly, the maximum of our function is . It is reached at the point .

We can say that the extrema of the function are equal to and .

Sometimes in tasks you need to find the largest and smallest values ​​of the function on a given segment. They do not necessarily coincide with extremes.

In our case smallest function value on the interval is equal to and coincides with the minimum of the function. But its largest value on this segment is equal to . It is reached at the left end of the segment.

In any case, the largest and smallest values ​​of a continuous function on a segment are achieved either at the extremum points or at the ends of the segment.