How to find the zeros of a function in a fraction. How to find zeros of a function

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

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

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

Examples.

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

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

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

Answer: x= -5.

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

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

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

Answer: x=3; x=4.

Instructions

1. The zero of a function is the value of the argument x at which the value of the function is equal to zero. However, only those arguments that are within the scope of the definition of the function under study can be zeros. That is, there are a lot of values ​​for which the function f(x) is useful. 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 a quadratic equation are calculated with support for finding the discriminant. 2x?+5x+2 = 0;D = b?-4ac = 5?-4*2*2 = 9;x1 = (-b+?D)/2*a = (-5+3)/2*2 = -0.5;x2 = (-b-?D)/2*a = (-5-3)/2*2 = -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. Find out the OOF, to do this, check the initial expression for the presence of even roots of the form?f (x), for the presence of fractions in the function with an argument in the denominator, for the presence of logarithmic or trigonometric expressions. 4. When considering a function with an expression under a root of an even degree, take as the domain of definition all the arguments x, the values ​​of which do not turn the radical expression into a negative number (on the contrary, the function does not make sense). Check whether the detected zeros of the function fall within a certain range of acceptable x values. 5. The denominator of the fraction cannot go to zero; therefore, exclude those arguments x that lead to such a result. For logarithmic quantities, only those values ​​of the argument should be considered for which the expression itself is greater than zero. Zeros of the function that turn the sublogarithmic expression into zero or a negative number must be discarded from the final result. Note! When finding the roots of an equation, extra roots may appear. This is easy to check: just substitute the resulting value of the argument into the function and make sure whether the function turns to zero. Helpful advice Occasionally a function is not expressed in an obvious way through its argument, then it is easy to know what this function is. An example of this is the equation of a circle.

Function zeros The abscissa value at which the value of the function is equal to zero is called.

If a function is given by its equation, then the zeros of the function will be the solutions to the equation. If a graph of a function is given, then the zeros of the function are the values ​​at which the graph intersects the x-axis.

Function is one of the most important mathematical concepts. Function - variable dependency at from 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 definition of the function. All values ​​that the dependent variable takes (variable y), form the range of values ​​of the function.

Function graph 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 ordinate axis y. To graph a function, you need to know the properties of the function. The main properties of the function will be discussed below!

To build a graph of a function, 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 they will help you solve problems in mathematics, chemistry, geometry, probability theory and many other subjects!

Basic properties of functions.

1) Function domain and function range.

The domain of a function is the set of all valid valid argument values x(variable x), for which the function y = f(x) determined.
The range of a function is the set of all real values y, which the function accepts.

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

2) Function zeros.

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

3) Intervals of constant sign of a function.

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

4) Monotonicity of the function.

An increasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a larger value of the function.

A decreasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a smaller value of the function.

5) Even (odd) function.

An even function is a function whose domain of definition is symmetrical 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 ordinate.

An odd function is a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality is true 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 is a positive number M such that |f(x)| ≤ M for all values ​​of x. If such a number does not exist, then the function is unlimited.

7) Periodicity of the function.

A function f(x) is periodic if there is a nonzero 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 a function, you can easily explore the function and, using the properties of the function, you can build a graph of the function. Also look at the material about the truth table, multiplication table, periodic table, table of derivatives and table of integrals.

Function zeros

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

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

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

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

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

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

Thus, 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, solve the quadratic equation

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

3) Find the zeros of the function

A fraction makes sense if the denominator is non-zero. Therefore, x²-1≠0, x² ≠ 1, x ≠±1. That is, the domain of definition of a given function (DO)

Of 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, you need to find the points of intersection of the function graph with the abscissa axis.

If the graph does not intersect 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

Function zeros rule

Basic concepts and properties of functions

Rule (law of) correspondence. Monotonic function .

Limited and unlimited functions. Continuous and

discontinuous functions . Even and odd functions.

Periodic function. Period of the function.

Function zeros . Asymptote .

The domain of definition and the range of values ​​of a function. In elementary mathematics, functions are studied only on the set of real numbers R . This means that the function argument can only take those real values ​​for which the function is defined, i.e. it also accepts only real values. A bunch of X all valid valid argument values x, for which the function y = f (x) is defined, called domain of the function. A bunch of Y all real values y, which 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 , according to which for each element from the set X you can find one and only one element from the set Y, is called a function .

From this definition it follows that a function is considered defined if:

— the domain of definition of the function is specified X ;

— the function range is specified Y ;

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

argument value, 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 shown in Fig. 3 is limited, but not monotonic. The function in Fig. 4 is just the opposite, monotonic, but unlimited. (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 when x = a, i.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 explosive at the point x = a .

If the function is continuous during everyone points of its domain of definition, then it is called continuous function.

Even and odd functions. If for any x from the domain of definition of the function the following holds: f (— x) = f (x), then the function is called even; if it happens: 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 with respect to the origin(Fig. 6).

Periodic function. Function f (x) — periodic, if such a thing exists non-zero number T what for any x from the domain of definition of the function the following holds: f (x + T) = f (x). This least the number is called period of the function. 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, addition 2 n not to the sine argument

changes its value e. Is there another number with this

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

valid for any value x. But then it has

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

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

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

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

we know that this is true only when P = 2 n. Since the smallest

non-zero number from 2 n is 2, then this number

and there is a period sin x. It can be proven in a similar way that 2

is also a period for cos x .

Prove that the functions tan x and cot x have period .

Example 2. What number is the period of the function sin 2 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 zeros. The argument value at which the function is equal to 0 is called zero ( root) function. A function may have multiple zeros. For example, the function y = x (x + 1) (x- 3) has three zeros: x = 0, x = — 1, x= 3. Geometrically null function – this is the abscissa of the point of intersection of the function graph with the axis X .

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

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

Topic 6. “Interval method.”

If f (x) f (x 0) for x x 0, then the function f (x) is called continuous at point 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 continuity interval of the function). The graph of a function on this interval is a continuous line, which they say can 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.

A method for solving inequalities with one variable, the interval method, is based on this property. 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 maintains 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 using the interval method.

Interval method for inequalities of the form

Interval method. Average level.

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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 dot is the zero of the function (). Let's show the signs of this function on the number axis:

We say that “the function changes sign when passing through the point”.

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

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

Quadratic function

I hope you remember how to solve quadratic inequalities? If not, read the topic “Quadratic Inequalities.” Let me remind you of 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 a 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 alternately change when passing through each root.

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

Recall that a square trinomial can be factorized:

Let's mark the roots on the axis:

We remember that the sign of a function can only change when passing through the root. Let's 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 at only one arbitrarily chosen point: at the remaining points of the interval the sign will be the same.

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

Well, when (substitute, for example), both brackets are negative, which means 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's also consider cases when the function has no zeros, or only one.

If they are not there, then there are no roots. This means that there will be no “passing through the root”. This means that the function takes only one sign on the entire number line. It can be easily determined 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 rule can we come up with for such situations?

If you factor such a function, you 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 highlight the root, when passing through which the sign does not change, by circling it with a square:

We will call such a root multiples.

Interval method 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 select intervals depending on the sign of the inequality. For example:

Let's measure the roots on the axis and place 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 sides of which are rational expressions (see “Rational Equations”).

Example:

All factors except one are “linear” here, that is, they contain a variable only to the first power. 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 this for yourself), and therefore does not affect the sign of the entire inequality. This means that we can divide the left and right sides of the inequality by 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 becomes zero, mark these points on the axis and arrange the signs. I would like to draw your attention to a very important fact:

In the case of an even number, we do the same 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 does not apply: the sign will still change when passing through the root. Therefore, we do not do anything additional with such a root, as if it were not a multiple. The above rules apply to all even and odd powers.

What should we write in the answer?

If the alternation of signs is violated, you need to be very careful, because if the inequality is not strict, the answer should include all shaded points. But some of them often stand apart, that is, they are not included in the shaded area. In this case, we add them to the answer as isolated points (in curly braces):

Examples (decide for yourself):

Answers:

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

2. Let's find the zeros of the function.

f(x) at x .

Answer f(x) at x .

2) x 2 >-4x-5;

x 2 +4x +5>0;

Let f(x)=x 2 +4x +5 then Let us find such x for which f(x)>0,

D=-4 No zeros.

4. Systems of inequalities. Inequalities and systems of inequalities with two variables

1) The set of solutions to a system of inequalities is the intersection of the sets of solutions to the inequalities included in it.

2) The set of solutions to the inequality f(x;y)>0 can be graphically depicted on the coordinate plane. Typically, the line defined by the equation f(x;y) = 0 divides the plane into 2 parts, one of which is the solution to the inequality. To determine which part, you need to substitute the coordinates of an arbitrary point M(x0;y0) that does not lie on the line f(x;y)=0 into the inequality. If f(x0;y0) > 0, then the solution to the inequality is the part of the plane containing the point M0. if f(x0;y0)<0, то другая часть плоскости.

3) The set of solutions to a system of inequalities is the intersection of the sets of solutions to the inequalities included in it. Let, for example, be given a system of inequalities:

.

For the first inequality, the set of solutions is a circle of radius 2 and centered at the origin, and for the second, it is a half-plane located above the straight line 2x+3y=0. The set of solutions of this system is the intersection of these sets, i.e. semicircle.

4) Example. Solve the system of inequalities:

The solution to the 1st inequality is the set , the 2nd is the set (2;7) and the third is the set .

The intersection of these sets is the interval (2;3], which is the set of solutions to the system of inequalities.

5. Solving rational inequalities using the interval method

The method of intervals is based on the following property of the binomial (x-a): the point x = α divides the number axis into two parts - to the right of the point α the binomial (x-α)>0, and to the left of the point α (x-α)<0.

Let it be necessary to solve the inequality (x-α 1)(x-α 2)...(x-α n)>0, where α 1, α 2 ...α n-1, α n are fixed numbers, among which there are no equals, and such that α 1< α 2 <...< α n-1 < α n . Для решения неравенства (x-α 1)(x-α 2)...(x‑α n)>0 using the interval method proceed as follows: the numbers α 1, α 2 ...α n-1, α n are plotted on the numerical axis; in the interval to the right of the largest of them, i.e. numbers α n, put a plus sign, in the interval following it from right to left put a minus sign, then a plus sign, then a minus sign, etc. Then the set of all solutions to the inequality (x-α 1)(x‑α 2)...(x-α n)>0 will be the union of all intervals in which the plus sign is placed, and the set of solutions to the inequality (x-α 1 )(x-α 2)...(x‑α n)<0 будет объединение всех промежутков, в которых поставлен знак «минус».

1) Solving rational inequalities (i.e. inequalities of the form P(x) Q(x) where are polynomials) is based on the following property of a continuous function: if a continuous function vanishes at points x1 and x2 (x1; x2) and has no other roots between these points, then in the intervals (x1; x2) the function retains its sign.

Therefore, to find intervals of constant sign of the function y=f(x) on the number line, mark all the points at which the function f(x) vanishes or suffers a discontinuity. These points divide the number line into several intervals, inside each of which the function f(x) is continuous and does not vanish, i.e. saves the sign. To determine this sign, it is enough to find the sign of the function at any point of the considered interval of the number line.

2) To determine intervals of constant sign of a rational function, i.e. To solve a rational inequality, we mark on the number line the roots of the numerator and the roots of the denominator, which are also the roots and breakpoints of the rational function.

Solving inequalities using the interval method

3. < 20.

Solution. The range of acceptable values ​​is determined by the system of inequalities:

For the function f(x) = – 20. Find f(x):

whence x = 29 and x = 13.

f(30) = – 20 = 0.3 > 0,

f(5) = – 1 – 20 = – 10< 0.

Answer: . Basic methods for solving rational equations. 1) The simplest: solved by the usual simplifications - reduction to a common denominator, reduction of similar terms, and so on. Quadratic equations ax2 + bx + c = 0 are solved by...

X changes on the interval (0,1], and decreases on the interval = ½ [
-(1/3)
], with | z|< 1.

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

With) f(z) = ½ [
]= - ½ [
] =

= - ½ = -
, with |2 - z| < 1

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

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

Etc. Investigate the convergence of the series

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

Solution. This 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 .