Zeros of the function according to the schedule. Function zero. Zero function rule

Function is one of the most important mathematical concepts. Function - variable dependency at from a variable x, if each value X matches a single value at. variable X called the independent variable or argument. variable at called the dependent variable. All values ​​of the independent variable (variable x) form the domain of the function. All values ​​that the dependent variable takes (variable y), form the range of the function.

Function Graph they call the set of all points of the coordinate plane, the abscissas of which are equal to the values ​​of the argument, and the ordinates are equal to the corresponding values ​​of the function, that is, the values ​​of the variable are plotted along the abscissa axis x, and the values ​​of the variable are plotted along the y-axis y. To plot a function, you need to know the properties of the function. The main properties of the function will be discussed below!

To plot a function graph, we recommend using our program - Graphing Functions Online. If you have any questions while studying the material on this page, you can always ask them on our forum. Also on the forum you will be helped to solve problems in mathematics, chemistry, geometry, probability theory and many other subjects!

Basic properties of functions.

1) Function scope and function range.

The scope of a function is the set of all valid values ​​of the argument x(variable x) for which the function y = f(x) defined.
The range of a function is the set of all real values y that the function accepts.

In elementary mathematics, functions are studied only on the set of real numbers.

2) Function zeros.

Zero of the function is the value of the argument at which the value of the function is equal to zero.

3) Intervals of sign constancy of a function.

The intervals of constant sign of a function are such sets of argument values ​​on which the values ​​of the function are only positive or only negative.

4) Monotonicity of the function.

Increasing function (in some interval) - a function in which a larger value of the argument from this interval corresponds to a larger value of the function.

Decreasing function (in some interval) - a function in which a larger value of the argument from this interval corresponds to a smaller value of the function.

5) Even (odd) functions.

An even function is a function whose domain of definition is symmetric with respect to the origin and for any X from the domain of definition the equality f(-x) = f(x). The graph of an even function is symmetrical about the y-axis.

An odd function is a function whose domain of definition is symmetric with respect to the origin and for any X from the domain of definition the equality f(-x) = - f(x). The graph of an odd function is symmetrical about the origin.

6) Limited and unlimited functions.

A function is called bounded if there exists a positive number M such that |f(x)| ≤ M for all values ​​of x . If there is no such number, then the function is unbounded.

7) Periodicity of the function.

A function f(x) is periodic if there exists a non-zero number T such that for any x f(x+T) = f(x). This smallest number is called the period of the function. All trigonometric functions are periodic. (Trigonometric formulas).

Having studied these properties of the function, you can easily explore the function and, using the properties of the function, you can plot the graph of the function. Also look at the material about the truth table, the multiplication table, the periodic table, the table of derivatives and the table of integrals.

Function zeros

What are function zeros? How to determine the zeros of a function analytically and graphically?

Function zeros are the values ​​of the argument at which the function is equal to zero.

To find the zeros of the function given by the formula y=f(x), we need to solve the equation f(x)=0.

If the equation has no roots, then the function has no zeros.

1) Find the zeros of the linear function y=3x+15.

To find the zeros of the function, we solve the equation 3x+15 =0.

So the zero of the function is y=3x+15 - x= -5 .

2) Find the zeros of the quadratic function f(x)=x²-7x+12.

To find the zeros of the function, we solve the quadratic equation

Its roots x1=3 and x2=4 are the zeros of this function.

3) Find the zeros of the function

A fraction makes sense if the denominator is different from zero. Therefore, x²-1≠0, x² ≠ 1,x ≠±1. That is, the domain of definition of this function (ODZ)

From the roots of the equation x²+5x+4=0 x1=-1 x2=-4, only x=-4 is included in the domain of definition.

To find the zeros of a function given graphically, it is necessary to find the points of intersection of the graph of the function with the x-axis.

If the graph does not cross the Ox axis, the function has no zeros.

the function whose graph is shown in the figure has four zeros -

In algebra, the problem of finding the zeros of a function occurs both as an independent task and when solving other problems, for example, when studying a function, solving inequalities, etc.

www.algebraclass.ru

Zero function rule

Basic concepts and properties of functions

rule (law of) conformity. Monotonic function .

Limited and unlimited functions. Continuous and

discontinuous function . Even and odd functions.

Periodic function. Function period.

Function zeros . Asymptote .

The scope and range of the function. In elementary mathematics, functions are studied only on the set of real numbers R . This means that the function argument can only take on those real values ​​for which the function is defined, i.e. e . it also only accepts real values. Lots of X all valid valid values ​​of the argument x, for which the function y = f (x) is defined, called function scope. Lots of Y all real values y that the function accepts is called function range. Now we can give a more precise definition of the function: rule (law of) correspondence between sets X And Y , by which for each element from the set X you can find one and only one element from the set Y, is called a function .

It follows from this definition that a function is considered given if:

- the scope of the function is set X ;

— range of function values ​​is set Y ;

- the rule (law) of correspondence is known, and such that for each

argument values, only one function value can be found.

This requirement of uniqueness of the function is mandatory.

monotonic function. If for any two values ​​of the argument x 1 and x 2 of the condition x 2 > x 1 follows f (x 2) > f (x 1), then the function f (x) is called increasing; if for any x 1 and x 2 of the condition x 2 > x 1 follows f (x 2)

The function depicted in Fig. 3 is bounded, but not monotonic. The function in Figure 4 is just the opposite, monotonic but unbounded. (Explain this please!)

Continuous and discontinuous functions. Function y = f (x) is called continuous at the point x = a, if:

1) the function is defined for x = a, i.e. e. f (a) exists;

2) exists finite limit lim f (x) ;

If at least one of these conditions is not met, then the function is called discontinuous at the point x = a .

If the function is continuous in all points of its domain of definition, then it is called continuous function.

Even and odd functions. If for any x from the scope of the function definition takes place: f (— x) = f (x), then the function is called even; if it does: f (— x) = — f (x), then the function is called odd. Graph of an even function symmetrical about the Y axis(Fig.5), a graph of an odd function Sim metric about the origin(Fig. 6).

Periodic function. Function f (x) — periodical if there is such non-zero number T what for any x from the scope of the function definition takes place: f (x + T) = f (x). Such least the number is called function period. All trigonometric functions are periodic.

EXAMPLE 1. Prove that sin x has a period of 2.

SOLUTION We know that sin ( x + 2 n) = sin x, where n= 0, ± 1, ± 2, …

Therefore, adding 2 n to the sine argument

changes its value e . Is there another number with this

Let's pretend that P- such a number, i.e. e. equality:

valid for any value x. But then it has

place and at x= / 2, i.e. e.

sin(/2 + P) = sin / 2 = 1.

But according to the reduction formula sin (/ 2 + P) = cos P. Then

it follows from the last two equalities that cos P= 1, but we

We know that this is only true for P = 2 n. Since the smallest

a non-zero number out of 2 n is 2 , then this number

and there is a period sin x. It is proved similarly that 2

is a period for cos x .

Prove that the functions tan x and cat x have a period.

Example 2. What number is the period of the sin 2 function x ?

Solution. Consider sin 2 x= sin(2 x + 2 n) = sin [ 2 ( x + n) ] .

We see that adding n to the argument x, does not change

function value. Smallest non-zero number

from n is , so this is the period sin 2 x .

Function nulls. The value of the argument for which the function is equal to 0 is called zero ( root) functions. A function can have multiple zeros. For example, the function y = x (x + 1) (x- 3) has three zeros: x = 0, x = — 1, x= 3. Geometrically function null – is the abscissa of the point of intersection of the graph of the function with the axis X .

Figure 7 shows the graph of the function with zeros: x = a , x = b And x = c .

Asymptote. If the graph of a function approaches a certain straight line indefinitely as it moves away from the origin, then this straight line is called asymptote.

Topic 6. "Method of intervals".

If f (x) f (x 0) for x x 0, then the function f (x) is called continuous at x 0.

If a function is continuous at every point of some interval I, then it is called continuous on the interval I (the interval I is called function continuity interval). The graph of the function on this interval is a continuous line, which is said to be "drawn without lifting the pencil from the paper."

Property of continuous functions.

If on the interval (a ; b) the function f is continuous and does not vanish, then it retains a constant sign on this interval.

The method of solving inequalities with one variable is based on this property - the method of intervals. Let the function f(x) be continuous on the interval I and vanish at a finite number of points in this interval. By the property of continuous functions, these points divide I into intervals, in each of which the continuous function f(x) c guards a constant sign. To determine this sign, it is enough to calculate the value of the function f(x) at any one point from each such interval. Based on this, we obtain the following algorithm for solving inequalities by the interval method.

The interval method for inequalities of the form

Find the domain of the function f(x) ;

Find the zeros of the function f(x) ;

Draw the domain of definition and zeros of the function on the number line. The zeros of a function divide its domain of definition into intervals, in each of which the function retains a constant sign;

Find the signs of the function in the obtained intervals by calculating the value of the function at any one point from each interval;

Write down the answer.

interval method. Average level.

Do you want to test your strength and find out the result of how ready you are for the Unified State Examination or the OGE?

Linear function

A function of the form is called linear. Let's take a function as an example. It is positive at 3″> and negative at. The point is the zero of the function (). Let's show the signs of this function on the real axis:

We say that "the function changes sign when passing through a point".

It can be seen that the signs of the function correspond to the position of the graph of the function: if the graph is above the axis, the sign is “ ”, if it is below it, “ ”.

If we generalize the resulting rule to an arbitrary linear function, we get the following algorithm:

We find the zero of the function;

We mark it on the numerical axis;

We determine the sign of the function on opposite sides of zero.

quadratic function

I hope you remember how quadratic inequalities are solved? If not, read the topic "Square inequalities". Let me remind you the general form of a quadratic function: .

Now let's remember what signs the quadratic function takes. Its graph is a parabola, and the function takes the sign “ ” for those in which the parabola is above the axis, and “ ” - if the parabola is below the axis:

If the function has zeros (values ​​at which), the parabola intersects the axis at two points - the roots of the corresponding quadratic equation. Thus, the axis is divided into three intervals, and the signs of the function change alternately when passing through each root.

Is it possible to somehow determine the signs without drawing a parabola each time?

Recall that the square trinomial can be factorized:

Note the roots on the axis:

We remember that the sign of a function can only change when passing through the root. We use this fact: for each of the three intervals into which the axis is divided by roots, it is enough to determine the sign of the function only at one arbitrarily chosen point: at the other points of the interval, the sign will be the same.

In our example: for 3″> both expressions in brackets are positive (we substitute, for example: 0″>). We put the sign "" on the axis:

Well, if (substitute, for example) both brackets are negative, then the product is positive:

That's what it is interval method: knowing the signs of the factors on each interval, we determine the sign of the entire product.

Let us also consider cases when the function has no zeros, or it is only one.

If there are none, then there are no roots. This means that there will be no “passage through the root”. This means that the function on the entire number axis takes only one sign. It is easy to determine by substituting it into a function.

If there is only one root, the parabola touches the axis, so the sign of the function does not change when passing through the root. What is the rule for such situations?

If we factor out such a function, we get two identical factors:

And any squared expression is non-negative! Therefore, the sign of the function does not change. In such cases, we will select the root, when passing through which the sign does not change, circling it with a square:

Such a root will be called multiple.

The method of intervals in inequalities

Now any quadratic inequality can be solved without drawing a parabola. It is enough just to place the signs of the quadratic function on the axis, and choose the intervals depending on the inequality sign. For example:

We measure the roots on the axis and arrange the signs:

We need the part of the axis with the sign ""; since the inequality is not strict, the roots themselves are also included in the solution:

Now consider a rational inequality - an inequality, both parts of which are rational expressions (see "Rational Equations").

Example:

All factors except one - - here are "linear", that is, they contain a variable only in the first degree. We need such linear factors to apply the interval method - the sign changes when passing through their roots. But the multiplier has no roots at all. This means that it is always positive (check it yourself), and therefore does not affect the sign of the entire inequality. This means that you can divide the left and right sides of the inequality into it, and thus get rid of it:

Now everything is the same as it was with quadratic inequalities: we determine at what points each of the factors vanishes, mark these points on the axis and arrange the signs. I draw your attention to a very important fact:

In the case of an even number, we proceed in the same way as before: we circle the point with a square and do not change the sign when passing through the root. But in the case of an odd number, this rule is not fulfilled: the sign will still change when passing through the root. Therefore, we do nothing additionally with such a root, as if it is not a multiple of us. The above rules apply to all even and odd powers.

What do we write in the answer?

If the alternation of signs is violated, you need to be very careful, because with non-strict inequality, the answer should include all filled points. But some of them often stand alone, that is, they do not enter the shaded area. In this case, we add them to the response as isolated dots (in curly braces):

Examples (decide for yourself):

Answers:

1. If among the factors it is simple - this is the root, because it can be represented as.
   .

where it takes on the value zero. For example, for a function given by the formula

is null because

.

Function zeros are also called function roots.

The concept of zeros of a function can be considered for any functions whose range contains zero or a zero element of the corresponding algebraic structure.

For a function of a real variable, zeros are the values ​​at which the graph of the function intersects the x-axis.

Finding the zeros of a function often requires the use of numerical methods (for example, Newton's method, gradient methods).

One of the unsolved mathematical problems is finding the zeros of the Riemann zeta function.

Polynomial root

see also

Literature

Wikimedia Foundation. 2010 .

See what "Function Zero" is in other dictionaries:

The point where a given function f(z) vanishes; thus, N. f. f (z) is the same as the roots of the equation f (z) = 0. For example, the points 0, π, π, 2π, 2π,... are the zeros of the sinz function. Zeros of an analytic function (See Analytic ... ...

Zero function, zero function... Spelling Dictionary

This term has other meanings, see Zero. It is necessary to move the content of this article to the article "Function Zero". You can help the project by consolidating the articles. If you need to discuss the advisability of merging, replace this ... Wikipedia

Or a C string (from the name of the C language) or an ASCIZ string (from the name of the assembler directive.asciz) is a way of representing strings in programming languages, in which an array of characters is used instead of introducing a special string type, and the end ... ... Wikipedia

In quantum field theory, the accepted (slang) name for the property of the vanishing of the renormalization factor of the coupling constant, where g0 is the bare coupling constant from the interaction Lagrangian, phys. coupling constant dressed by interaction. Equality Z... Physical Encyclopedia

Null mutation n-allele- Null mutation, n. allele * null mutation, n. allele * null mutation or n. allele or silent a. a mutation leading to a complete loss of function in the DNA sequence in which it occurred ... Genetics. encyclopedic Dictionary

The statement in probability theory that any event (the so-called residual event), the occurrence of which is determined only by arbitrarily remote elements of a sequence of independent random events or random variables, has ... ... Mathematical Encyclopedia

1) A number that has the property that any (real or complex) number does not change when added to it. It is denoted by the symbol 0. The product of any number by N. is equal to N.: If the product of two numbers is equal to N., then one of the factors ... Mathematical Encyclopedia

Functions given by relations between independent variables not resolved with respect to the latter; these relations are one of the ways to define a function. For example, the relation x2 + y2 1 = 0 defines the N. f. … Great Soviet Encyclopedia

Argument values z under which f(z) goes to zero called. zero point, i.e. if f(a) = 0 , then a - zero point.

Def. Dot but called order zeron , if FKP can be represented in the form f(z) = , where
analytic function and
0.

In this case, in the expansion of the function in a Taylor series (43), the first n coefficients are zero

= =

Etc. Determine the order of zero for
and (1-cos z) at z = 0

=
=

zero 1st order

1 - cos z =
=

zero 2nd order

Def. Dot z =
called point at infinity And zero functions f(z), if f(
) = 0. Such a function expands into a series in negative powers z : f(z) =
. If first n coefficients are equal to zero, then we arrive at zero order n at a point at infinity: f(z) = z - n
.

Isolated singular points are divided into: a) removable singular points; b) order polesn; in) essential singular points.

Dot but called removable singular point functions f(z) if z
a lim f(z) = from - finite number .

Dot but called pole of ordern (n 1) features f(z) if the inverse function
= 1/ f(z) has order zero n at the point but. Such a function can always be represented as f(z) =
, where
- analytical function and
.

Dot but called essential point functions f(z), if z
a lim f(z) does not exist.

Laurent series

Consider the case of an annular convergence region r < | z 0 – a| < R centered on a point but for function f(z). We introduce two new circles L 1 (r) And L 2 (R) near the boundaries of the ring with a dot z 0 between them. Let us make a section of the ring, join the circles along the edges of the section, pass to a simply connected region, and in

Cauchy integral formula (39) we obtain two integrals over the variable z

f(z 0) =
+
, (42)

where integration goes in opposite directions.

For the integral over L 1 the condition | z 0 – a | > | z – a |, and for the integral over L 2 reverse condition | z 0 – a | < | z – a |. Therefore, the factor 1/( z – z 0) expand in a series (a) in the integral over L 2 and in series (b) in the integral over L one . As a result, we get the decomposition f(z) in the annular region in Laurent series in positive and negative powers ( z 0 – a)

f(z 0) =
A n (z 0 – a) n (43)

where A n =
=
;A -n =

Expansion in positive powers (z 0 - but) called right part Laurent series (Taylor series), and the expansion in negative powers is called. main part Laurent row.

If inside the circle L 1 there are no singular points and the function is analytic, then in (44) the first integral is equal to zero by the Cauchy theorem, and only the correct part remains in the expansion of the function. Negative powers in expansion (45) appear only when analyticity is violated within the inner circle and serve to describe the function near isolated singular points.

To construct the Laurent series (45) for f(z) one can calculate the expansion coefficients according to the general formula or use the expansions of the elementary functions included in f(z).

Number of terms ( n) of the main part of the Laurent series depends on the type of singular point: removable singular point (n = 0) ; essential singular point (n
); polen-th order(n - end number).

and for f(z) = dot z = 0 removable singular point, because there is no main part. f(z) = (z -
) = 1 -

b) For f(z) = dot z = 0 - 1st order pole

f(z) = (z -
) = -

c) For f(z) = e 1 / z dot z = 0 - essential singular point

f(z) = e 1 / z =

If f(z) is analytic in the domain D with the exception of m isolated singular points and | z 1 | < |z 2 | < . . . < |z m| , then when expanding the function in powers z the whole plane is divided into m+ 1 ring | z i | < | z | < | z i+ 1 | and the Laurent series has a different form for each ring. When expanding in powers ( z – z i ) the convergence region of the Laurent series is the circle | z – z i | < r, where r is the distance to the nearest singular point.

Etc. Expand the function f(z) =in Laurent's series in powers z And ( z - 1).

Solution. We represent the function in the form f(z) = - z 2 . We use the formula for the sum of a geometric progression
. In the circle |z|< 1 ряд сходится и f(z) = - z 2 (1 + z + z 2 + z 3 + z 4 + . . .) = - z 2 - z 3 - z 4 - . . . , i.e. decomposition contains only correct part. Let's move to the outer region of the circle |z| > 1 . We represent the function in the form
, where 1/| z| < 1, и получим разложение f(z) = z
=z + 1 +

Because , expansion of the function in powers ( z - 1) looks like f(z) = (z - 1) -1 + 2 + (z - 1) for everyone
1.

Etc. Expand the function in a Laurent series f(z) =
: a) in degrees z in a circle | z| < 1; b) по степеням z ring 1< |z| < 3 ; c) по степеням (z – 2).Decision. Let's decompose the function into simple fractions
= =+=
. From conditions z =1
A = -1/2 , z =3
B = ½.

but) f(z) = ½ [
] = ½ [
-(1/3)
], when | z|< 1.

b) f(z) = - ½ [
+
] = - (
), at 1< |z| < 3.

from) f(z) = ½ [
]= - ½ [
] =

= - ½ = -
, for |2 - z| < 1

It is a circle of radius 1 centered on a point z = 2 .

In some cases, power series can be reduced to a set of geometric progressions, and then it is easy to determine the area of ​​their convergence.

Etc. Investigate the convergence of a series

. . . + + + + 1 + () + () 2 + () 3 + . . .

Solution. It is the sum of two geometric progressions with q 1 = , q 2 = () . From the conditions of their convergence it follows < 1 , < 1 или |z| > 1 , |z| < 2 , т.е. область сходимости ряда кольцо 1 < |z| < 2 .

What are function zeros? The answer is quite simple - this is a mathematical term, which means the domain of a given function, on which its value is zero. Function zeros are also called Function zeros The easiest way to explain what function zeros are is with a few simple examples.

Examples

Consider a simple equation y=x+3. Since the zero of the function is the value of the argument at which y became zero, we substitute 0 on the left side of the equation:

In this case, -3 is the desired zero. For a given function, there is only one root of the equation, but this is not always the case.

Consider another example:

Substitute 0 on the left side of the equation, as in the previous example:

Obviously, in this case, there will be two zeros of the function: x=3 and x=-3. If the equation had an argument of the third degree, there would be three zeros. We can make a simple conclusion that the number of roots of the polynomial corresponds to the maximum degree of the argument in the equation. However, many functions, for example y=x 3 , at first sight contradict this statement. Logic and common sense suggest that this function has only one zero - at the point x=0. But in fact there are three roots, they just all coincide. If you solve the equation in complex form, this becomes obvious. x=0 in this case, the root, the multiplicity of which is 3. In the previous example, the zeros did not match, therefore they had a multiplicity of 1.

Definition algorithm

From the presented examples it is clear how to determine the zeros of the function. The algorithm is always the same:

1. Write a function.
2. Substitute y or f(x)=0.
3. Solve the resulting equation.

The complexity of the last item depends on the degree of the argument of the equation. When solving equations of high degrees, it is especially important to remember that the number of roots of the equation is equal to the maximum power of the argument. This is especially true for trigonometric equations, where dividing both parts by sine or cosine leads to the loss of roots.

Arbitrary degree equations are most easily solved by Horner's method, which was developed specifically for finding the zeros of an arbitrary polynomial.

The value of zeros of functions can be both negative and positive, real or lying in the complex plane, single or multiple. Or there may be no roots of the equation. For example, the function y=8 will not become zero for any x, because it does not depend on this variable.

The equation y=x 2 -16 has two roots, and both lie in the complex plane: x 1 =4i, x 2 =-4i.

Common Mistakes

A common mistake made by schoolchildren who have not yet really figured out what the zeros of a function are is replacing the argument (x) with zero, and not the value (y) of the function. They confidently substitute x = 0 into the equation and, based on this, find y. But this is the wrong approach.

Another error, as already mentioned, is the reduction by sine or cosine in the trigonometric equation, which is why one or more zeros of the function are lost. This does not mean that nothing can be reduced in such equations, but these "lost" factors must be taken into account in further calculations.

Graphical representation

You can understand what the zeros of a function are with the help of mathematical programs such as Maple. In it, you can build a graph by specifying the desired number of points and the desired scale. Those points at which the graph crosses the OX axis are the desired zeros. This is one of the fastest ways to find the roots of a polynomial, especially if its order is higher than the third. So if there is a need to regularly perform mathematical calculations, find the roots of polynomials of arbitrary degrees, build graphs, Maple or a similar program will be simply indispensable for performing and verifying calculations.

The mathematical representation of a function clearly shows how one quantity completely determines the value of another quantity. Traditionally, numerical functions are considered that associate one number with another. The zero of a function is usually called the value of the argument at which the function vanishes.

Instruction

1. In order to find the zeros of a function, you need to equate its right side to zero and solve the resulting equation. Imagine you are given a function f(x)=x-5.

2. To find the zeros of this function, let's take and equate its right side to zero: x-5=0.

3. Solving this equation, we get that x=5 and this value of the argument will be the zero of the function. That is, when the value of the argument is 5, the function f(x) vanishes.

Under submission functions in mathematics understand the relationship between the elements of sets. To put it more correctly, this is a “law” according to which an entire element of one set (called the domain of definition) is assigned a certain element of another set (called the domain of values).

You will need

• Knowledge in algebra and mathematical review.

Instruction

1. Values functions this is a certain area from which the function can take values. Let's say the range functions f(x)=|x| from 0 to infinity. To discover meaning functions at a certain point, you need to substitute for the argument functions its numerical equivalent, the resulting number will be meaning m functions. Let the function f(x)=|x| – 10 + 4x. discover meaning functions at the point x=-2. Substitute the number -2 instead of x: f(-2)=|-2| – 10 + 4*(-2) = 2 – 10 – 8 = -16. I.e meaning functions at point -2 is -16.

Note!
Before looking for the value of a function at a point, make sure that it is included in the scope of the function.

Useful advice
By a similar method it is possible to find the value of the function of several arguments. The difference is that instead of one number, you will need to substitute several - according to the number of arguments of the function.

The function is the established relation of the variable y from the variable x. Moreover, the entire value of x, called the argument, corresponds to the exceptional value of y - the function. In graphical form, the function is displayed on a Cartesian coordinate system in the form of a graph. The intersection points of the graph with the abscissa axis, on which the arguments x are plotted, are called the zeros of the function. Finding valid zeros is one of the tasks of finding a given function. In this case, all admissible values ​​of the independent variable x, which form the domain of the function definition (OOF), are taken into account.

Instruction

1. The zero of the function is the value of the argument x, at which the value of the function is zero. However, only those arguments that are included in the domain of definition of the function under study can be zeros. That is, in such a lot of values ​​\u200b\u200bfor which the function f (x) has a sense.

2. Write down the given function and equate it to zero, say f(x) = 2x?+5x+2 = 0. Solve the resulting equation and find its real roots. The roots of the quadratic equation are calculated with the support of finding the discriminant. 2x? + 5x + 2 \u003d 0; D \u003d b? -4ac \u003d 5? -4 * 2 * 2 \u003d 9; x1 \u003d (-b +? D) / 2 * a \u003d (-5 + 3) / 2 * 2 \u003d -0.5; x2 \u003d (-b-? D) / 2 * a \u003d (-5-3) / 2 * 2 \u003d -2. Thus, in this case, two roots of the quadratic equation are obtained, corresponding to the arguments of the initial function f(x).

3. Check all detected x values ​​for belonging to the domain of definition of the given function. Detect OOF, to do this, check the initial expression for the presence of roots of an even degree of the form? f (x), for the presence of fractions in a function with an argument in the denominator, for the presence of logarithmic or trigonometric expressions.

4. When considering a function with an expression under the root of an even degree, take for the domain of definition all the arguments x, the values ​​of which do not turn the root expression into a negative number (on the contrary, the function does not make sense). Specify whether the detected zeros of the function fall within a certain range of acceptable x values.

5. The denominator of a fraction cannot vanish, therefore exclude those arguments x that lead to such a result. For logarithmic quantities, one should consider only those values ​​of the argument for which the expression itself is larger than zero. Function zeros that turn the sublogarithmic expression to zero or a negative number must be discarded from the final result.

Note!
When finding the roots of an equation, extra roots may appear. It is easy to check this: it is enough to substitute the obtained value of the argument into the function and make sure that the function vanishes.

Useful advice
Occasionally a function is not explicitly expressed through its argument, then it is easy to know what the function is. An example of this is the equation of a circle.