Functional graphical method for solving exponential equations. Functional-graphical method for solving equations
In a standard school mathematics course, the properties of functions are used mainly to construct their graphs. The functional method for solving equations is used if and only if the equation F(x) = G(x) as a result of transformations or replacement of variables cannot be reduced to one or another standard equation that has a specific solution algorithm.
Unlike the graphical method, knowledge of the properties of functions allows you to find the exact roots of the equation, without the need to construct graphs of functions. Using the properties of functions helps to rationalize the solution of equations.
The following properties of the function are considered in the work: domain of definition of the function; function range; properties of monotonicity of a function; properties of convexity of a function; properties of even and odd functions.
Purpose of the work: to carry out some classification of non-standard equations using the general properties of functions, describe the essence of each property, give recommendations for its use, instructions for use.
All work is accompanied by the solution of specific problems proposed at the Unified State Examination in various years.
Chapter 1. Using the concept of domain of definition of a function.
Let's introduce a few key definitions.
The domain of definition of the function y = f(x) is the set of values of the variable x for which the function makes sense.
Let the equation f(x) = g(x) be given, where f(x) and g(x) are elementary functions defined on the sets D1, D2. Then the region D of admissible values of the equation will be a set consisting of those values of x that belong to both sets, that is, D = D1∩ D2. It is clear that when the set D is empty (D= ∅), then the equation has no solutions. (Appendix No. 1).
1. arcsin (x+2) +2x- x2 = x-2.
ODZ:-1 =0⇔-3
Answer: there are no solutions.
2. (x2-4x+3 +1)log5x5 + 1x(8x-2x2-6 + 1) = 0.
ODZ: x2-4x+3>=0,x>0.8x-2x2-6>=0⇔x∈(-infinity;1∪ 3;infinity),x>01
Check: x = 1.
(1-4+3 +1)log515 + (8-2-6 + 1) = 0,
0 = 0 - true.
x = 3. (9-12+3+1)log535 +13(24-18-6+1) = 0, log535 +13 = 0 - incorrect.
It often turns out to be sufficient to consider not the entire domain of definition of a function, but only its subset, on which the function takes values that satisfy certain conditions (for example, only non-negative values).
1. x+27-x(x-9 +1) = 1.
ODZ: x-9>=0, x>=9.
For x>=9 x+2>0, 7-x 0, thus, the product of the three factors on the left side of the equation is negative, and the right side of the equation is positive, which means the equation has no solutions.
Answer: ∅.
2. 3-x2+ x+2 = x-2.
ODZ: 3-x2>=0,x+2>=0,⇔ 3-x(3+x)>=0,x>=-2,⇔ -3=-2,⇔
On the set of admissible values, the left side of the equation is positive, and the right side is negative, which means that the equation has no solutions.
Answer: there are no solutions.
Chapter 2. Using the concept of function range.
The range of values of the function y = f(x) is the set of values of the variable y for acceptable values of the variable x.
A function y = f(x) is said to be bounded below (resp. above) on the set X if there exists a number M such that the inequality fx>=M holds on X (resp. fx
A function y = f(x) is called bounded on a given interval (contained in its domain of definition) if there is a number M >0 such that for all values of the argument belonging to this interval the inequality f(x) holds
Let the equation f(x) = g(x) be given, where g(x) are elementary functions defined on the sets D1, D2. Let us denote the range of variation of these functions as E1 and E2, respectively. If x1 is a solution to the equation, then the numerical equality f(x1) = g(x1) will hold, where f(x1) is the value of the function f(x) at x = x1, and g(x1) is the value of the function g(x) at x = x1. This means that if the equation has a solution, then the ranges of functions f(x) and g(x) have common elements (E1∩E2 !=∅). If the sets E1 and E2 do not contain such common elements, then the equation has no solutions.
Basic inequalities are used to evaluate expressions. (Appendix No. 2).
Let the equation f(x) = g(x) be given. If f(x)>=0 and g(x)
1. x2+2xsinxy+1=0.
Solution. There is a unit on the left side, which means we can use the basic trigonometric identity: x2+ 2xsinxy+ sin2xy+cos2xy=0.
The sum of the first three terms is a perfect square:
(x+sinxy)2+cos2xy =0.
Consequently, on the left side is the sum of squares; it is equal to zero when the expressions in the squares are simultaneously equal to zero. Let's write the system: cosxy=0,x+sinxy=0.
If cosxy=0, then sinxy= +-1, therefore this system is equivalent to a combination of two systems: x+1=0,cosxy=0 or x-1=0,cosxy=0.
Their solutions are pairs of numbers x=1, y = PI 2 + PIm, m∈Z, and x=-1, y = PI 2 + PIm, m∈Z.
Answer: x=1, y = PI 2 + PIm, m∈Z, and x=-1, y = PI 2 + PIm, m∈Z.
If on the interval X the largest value of one of the functions y = f(x), y = g(x) is equal to A and the smallest value of the other function is also equal to A, then the equation f(x) = g(x) is equivalent on the interval X to the system of equations fx=A,gx=A.
1. Find all values of a for which the equation has a solution
2cos222x-x2=a+3sin(22x-x2+1).
After replacing t= 22x-x2 we arrive at the equation cos(2t+PI3)=a-12.
The function t=2m increases, which means it reaches its greatest value at the highest value of m. But m=2х - x has the greatest value equal to 1. Then tmax = 22·1-1=2. Thus, the set of values of the function t = 22x-x2 is the interval (0;2, and the function cos(2t+PI3) is the interval -1;0.5). Consequently, the original equation has a solution for those and only those values of a that satisfy the inequalities -1Answer: -12. Solve the equation (log23)x+a+2 = (log94)x2+a2-6a-5.
Using the obvious inequalities
Answer: x= - 5+32 if a=1+32 and x=-5+32 if a= 1-32.
You can consider other equations in more detail. (Appendix No. 3).
Chapter 3. Using the property of monotonicity of a function.
A function y = f(x) is said to be increasing (respectively, decreasing) on a set X if on this set, as the argument increases, the values of the function increase (respectively, decrease).
In other words, the function y = f(x) increases on the set X if from x1∈X, x2∈X and x1 It decreases on this set if from x1∈X, x2∈X and x1 f(x2).
A function y = f(x) is said to be non-strictly increasing (respectively, non-strictly decreasing) on X if x1∈X, x2∈X and x1=f(x2)).
Functions that increase and decrease on X are called monotone on X, and functions that are not strictly increasing or decreasing on X are called non-strictly monotone on X.
To prove the monotonicity of functions, the following statements are used:
1. If a function f increases on a set X, then for any number C the function f + C also increases on X.
2. If the function f increases on the set X and C > 0, then the function Cf also increases on X.
3. If a function f increases on a set X, then the function - f decreases on this set.
4. If a function f increases on the set X and maintains sign on the set X, then the function 1f decreases on this set.
5. If functions f and g increase on a set X, then their sum f + g also increases on this set.
6. If the functions f and g are increasing and non-negative on the set X, then their product fg is also increasing on X.
7. If the function f is increasing and non-negative on the set X and n is a natural number, then the function fn is also increasing on X.
8. If both functions f(x) and g(x) are increasing or both are decreasing, then the function h(x) = f(g(x)) is an increasing function. If one of the functions is increasing. And the other is decreasing, then h(x) = f(g(x)) is a decreasing function.
Let us formulate theorems about equations.
Theorem 1.
If the function f(x) is monotonic on the interval X, then the equation f(x) = C has at most one root on the interval X.
Theorem 2.
If the function f(x) is monotonic on the interval X, then the equation f(g(x)) = f(h(x)) is equivalent on the interval X to the equation g(x) = h(x).
Theorem 3.
If the function f(x) increases on the interval X, and g(x) decreases on the interval X, then the equation g(x) = f(x) has at most one root on the interval X.
Theorem 4.
If the function f(x) increases on the interval X, then the equation f(f(x)) = x is equivalent on the interval X to the equation f(x) = x.
1. Find all values of a for which the equation has exactly three roots
4-x-alog3(x2-2x+3)+2-x2+2xlog13(2x-a+2)=0.
Solution. Let's transform this equation to the form
2x2-2xlog3(x2-2x+3)= 22x-a-1log3(2x-a+2).
If we put u = x2-2x, v=2x-a-1, then we arrive at the equation
2ulog3(u+3)= 2vlog3(v+3).
The function f (t) = 2tlog3(t+3) increases monotonically for t >-2, so from the last equation we can go to the equivalent u = v, x2-2x = 2x-a-1⇔(x-1)2=2x -a.
This equation, as can be seen from the figure, has exactly three roots in the following cases:
1. The vertex of the graph of the function y = 2x-a is located at the vertex of the parabola y = (x-1)2, which corresponds to a = 1;
2. The left ray of the graph y = 2x-a touches the parabola, and the right one intersects it at two points; this is possible with a=12;
3. The right ray touches and the left ray intersects the parabola, which occurs when a=32.
Let us explain the second case. The equation of the left ray is y = 2a-2x, its slope is -2. Therefore, the angular coefficient of the tangent to the parabola is equal to
2(x -1) = -2 ⇒ x = 0 and the tangent point has coordinates (0; 1). From the condition that this point belongs to the ray, we find a=12.
The third case can be considered similarly or using symmetry considerations.
Answer: 0.5; 1;1.5.
We can consider other equations in more detail. (Appendix No. 4).
Chapter 4. Using the properties of convexity.
Let a function f(x) be defined on an interval X, it is called strictly convex downward (upward) on X if for any u and v from X, u!=v and 0
Geometrically, this means that any point of the chord BC (that is, a segment with ends at points B(u;f(u)) and C(v;f(v)), different from points B and C, lies above (below) the point And the graph of the function f(x), corresponding to the same argument value (Appendix No. 5).
Functions that are strictly convex up and down are called strictly convex.
The following statements are true.
Theorem 1.
Let the function f(x) be strictly downward convex on the interval X, u ,v ∈X, u
The following statement follows from Theorem 1.
Theorem 2.
If the function f(x) is strictly convex on the interval X, the functions u = u(x), v = v(x), u1=u1(x), v1 = v1(x) are such that for all x from the ODZ equations f(u)+f(v) = f(u1) + f(v1) (1) their values u(x), v(x), u1(x), v1(x) are contained in X and the condition u is satisfied +v = u1 +v1, then the equation f(u)+f(v) = f(u1) + f(v1) (2) on the ODZ is equivalent to the set of equations u (x) = u1(x), u(x) = v1(x) (3).
1. 41-sin4x+41-cos4x=412.
Solution. If we set fx= 41-x2, u=cos2x, v=sin2x, u1=v1=12, then this equation will be written in the form (1). Since f"x= -x24(1-x2)3, f""x=-2+x244(1-x2)7, then the function fx is strictly convex upward on the segment -1;1. Obviously, the remaining conditions are satisfied Theorem 2 and, therefore, the equation is equivalent to the equation cos2x = 0.5, x = PI4 +PIk2, where k∈Z.
Answer: x = PI4 +PIk2, where k∈Z.
Theorem 3.
Let the function fx be strictly convex on the interval X and u,v, λv+(1-λ)u∈X. Then the equality f (λv+(1-λ)u) = λf(v)+(1-λ)f(u) (4) is valid if and only if either u=v or λ=0, or λ=1.
Examples: sin2xcos3x+cos2xsin3x∙1+sin2xcos3x+cos2xsin3x= sin2xcos3x1+cos3x+cos2xsin3x1+sin3x.
The equation has the form (4) if fx=x1+x= x+x2, u=sin3x, v= cos3x, λ=sin2x.
It is obvious that the function fx is strictly convex downward on R. Therefore, by Theorem 3, the original equation is equivalent to the set of equations sinx=0, sin2x=1, cos3x=sin3x.
From here we get that its solutions will be PIk2, PI12+PIn3, where k,n∈Z.
Answer: PIk2, PI12+PIn3, where k,n∈Z.
The use of convexity properties is also used to solve more complex equations. (Appendix No. 6).
Chapter 5. Using the even or odd properties of functions.
A function fx is called even if for any value x taken from the domain of definition of the function, the value - x also belongs to the domain of definition and the equality f-x = fx holds. A function fx is called odd if for any value x taken from the domain of definition of the function, the value - x also belongs to the domain of definition and the equality f-x = - fx holds.
From the definition it follows that the domains of even and odd functions are symmetrical about zero (a necessary condition).
For any two symmetric values of the argument from the domain of definition, the even function takes equal numerical values, and the odd one - equal in absolute value, but of opposite sign.
Theorem 1.
The sum, difference, product and quotient of two even functions are even functions.
Theorem 2.
The product and quotient of two odd functions are even functions.
Let us have the equation F(x)=0, where F(x) is an even or odd function.
To solve the equation F(x) = 0, where F(x) is an even or odd function, it is enough to find positive (or negative) roots symmetrical to those obtained, and for an odd function the root will be x = 0 if this value is within the domain of definition F(x). For an even function, the value x = 0 is checked by direct substitution into the equation.
We have even functions on both sides of the equation. Therefore, it is enough to find solutions for x>=0. Since x=0 is not a root of the equation, consider two intervals: (0;2, 2;infinity.
a) On the interval (0;2 we have:
8x= 2x+2-x+2, 23x=24, x= 43.
b) On the interval 2;infinity we have:
8x= 2x+2+x-2.23x=22x, x=0.
But since x = 0 is not a root of the equation, then for x>0 this equation has a root x = 43. Then x = - 43 is also a root of the equation.
Answer: 43; - 43.
The author believes that the work can be used by teachers and students of general education in elective classes, in preparation for mathematical Olympiads, passing the Unified State Exam, and entrance exams to technical educational institutions.
The accuracy of such a solution is low, but with the help of a graph you can intelligently choose the first approximation from which to begin further solving the equation. There are two ways to solve equations graphically.
First way . All terms of the equation are transferred to the left side, i.e. the equation is presented in the form f(x) = 0. After this, a graph of the function y = f(x) is constructed, where f(x) is the left side of the equation. Abscissas of the points of intersection of the graph of the function y = f(x) with the axis Ox and are the roots of the equation, because at these points y = 0.
Second way . All terms of the equation are divided into two groups, one of them is written on the left side of the equation, and the other on the right, i.e. represent it in the form j(x) = g(x). After this, graphs of two functions y = j(x) and y = g(x) are plotted. The abscissas of the intersection points of the graphs of these two functions serve as the roots of this equation. Let the point of intersection of the graphs have an abscissa x o, the ordinates of both graphs at this point are equal to each other, i.e. j(x o) = g(x o). From this equality it follows that x 0 is the root of the equation.
Root separation
The process of finding approximate values of the roots of the equation is divided into two stages:
1) separation of roots;
2) refinement of the roots to a given accuracy.
The x root of the equation f(x) = 0 is considered separated on the interval if the equation f(x) = 0 has no other roots on this interval.
Separating roots means dividing the entire range of acceptable values into segments, each of which contains one root.
Graphic method of root separation - in this case, proceed in the same way as with the graphical method of solving equations.
If the curve touches the x-axis, then at this point the equation has a double root (for example, the equation x 3 - 3x + 2 = 0 has three roots: x 1 = -2; x 2 = x 3 = 1).
If the equation has a threefold real root, then at the point of contact with the axis X the curve y = f(x) has an inflection point (for example, the equation x 3 - 3x 2 + 3x - 1 = 0 has a root x 1 = x 2 = x 3 = 1).
Analytical root separation method . To do this, use some properties of functions.
Theorem 1 . If the function f(x) is continuous on a segment and takes values of different signs at the ends of this segment, then inside the segment there is at least one root of the equation f(x) = 0.
Theorem 2. If the function f(x) is continuous and monotonic on a segment and takes values of different signs at the ends of the segment, then the segment contains the root of the equation f(x) = 0, and this root is unique.
Theorem 3 . If the function f(x) is continuous on a segment and takes values of different signs at the ends of this segment, and the derivative f "(x) maintains a constant sign inside the segment, then inside the segment there is a root of the equation f(x) = 0 and, moreover, a unique one.
If the function f(x) is given analytically, then domain of existence (domain of definition) of the function is the set of all those real values of the argument for which the analytical expression defining the function does not lose its numerical meaning and takes only real values.
The function y = f(x) is called increasing , if as the argument increases, the value of the function increases, and decreasing , if as the argument increases, the value of the function decreases.
The function is called monotonous , if in a given interval it either only increases or only decreases.
Let the function f(x) be continuous on the segment and take values of different signs at the ends of the segment, and the derivative f "(x) maintains a constant sign on the interval. Then if at all points of the interval the first derivative is positive, i.e. f "(x) >0, then the function f(x) in this interval increases . If at all points of the interval the first derivative is negative, i.e. f "(x)<0, то функция в этом интервале decreases .
Let the function f(x) on an interval have a second-order derivative that maintains a constant sign throughout the entire interval. Then if f ""(x)>0, then the graph of the function is convex down ; if f ""(x)<0, то график функции является convex up .
Points at which the first derivative of a function is equal to zero, as well as those at which it does not exist (for example, it turns to infinity), but the function maintains continuity, are called critical .
Procedure for separating roots using the analytical method:
1) Find f "(x) - the first derivative.
2) Make a table of signs of the function f(x), assuming X equal to:
a) critical values (roots) of the derivative or those closest to them;
b) boundary values (based on the range of permissible values of the unknown).
Example. Separate the roots of the equation 2 x - 5x - 3 = 0.
We have f(x) = 2 x - 5x - 3 . The domain of definition of the function f(x) is the entire numerical axis.
Let's calculate the first derivative f "(x) = 2 x ln(2) - 5.
We equate this derivative to zero:
2 x log(2) - 5 = 0 ; 2 x log(2) = 5 ; 2 x = 5/ln(2) ; xlg(2) = lg(5) - lg(ln(2)) .
We compile a table of signs of the function f(x), assuming X equal to: a) critical values (roots of the derivative) or closest to them; b) boundary values (based on the range of permissible values of the unknown):
The roots of the equation lie in the intervals (-1.0) and (4.5).
Functional-graphical method for solving the inequality f(x)< g(x). 1. Подбором найдем корень уравнения f(x)=g(x), используя свойства монотонных функций; 2. Построим схематически графики обеих функций, проходящие через точку с найденной абсциссой; 3. Выберем решение неравенства, соответствующее знаку неравенства; 4. Запишем ответ. :
Slide 9 from the presentation "Exponential Equations and Inequalities". The size of the archive with the presentation is 174 KB.Algebra 11th grade
summary of other presentations“Equations of the third degree” - (1). Tartaglia refuses. On February 12, Cardano repeats his request. "Great Art" X3 + px + q = 0. Example: x3 – 5 x2 + 8 x – 4 = 0 x3 – 2 x2 –3 x2 + 8x – 4 = 0 x2 (x – 2) – (3 x2 – 8x + 4) = 0 3 x2 – 8x + 4 = 0 x = 2 x = 2/3 x2 (x – 2) – (3 (x –2) (x – 2/3)) = 0 x2 (x – 2) – (( x – 2) (3x – 2)) = 0 (x – 2)(x2 – 3x + 2) = 0 x – 2 = 0 x2 – 3x + 2 = 0 x = 2 x = 2 x = 1 Answer: x = 2; x = 1. Our formula gives: Municipal educational institution “Secondary school No. 24”. X3 + ax = b (1). Here p = 6 and q = -2. First example:
“Application of the definite integral” - Ch. 4. Development of an elective on the topic “Definite Integral”. Definite integral. §4. Properties of a definite integral. Ch. 2. Various approaches to integral theory in textbooks for schoolchildren. §1. Volume of a body of revolution. §6. Introduction. Darboux sums. §3. Mechanical work. Goal: Approaches to the construction of integral theory: Introductory remarks. §2. Integration methods. §3. Conclusion. Chapter 3. Application of a definite integral. §1.