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Part 1: Primary school mathematics teachers recruiting professional knowledge test questions compilation teacher one, single-choice questions.

1. Among the following conditions, it can be determined that the quadrilateral is a parallelogram ()A. A set of diagonals is equal to B, and two diagonal lines are equally divided by each other. C. A pair of opposite sides is equal to D. Two diagonal lines Vertically 3, the monotonic reduction interval of the function y=6x3-12x2+6x+1 is ()A.(,)B.(,1)C.(1,+)D.(-1,-)4,( ) is the Newton-Lebrecht formula, where F(x) is an original function of f(x).

(x)dxF(a)F(b)(x)dxF(a)F(b) If the radii of the two circles are 1cm and 5cm, respectively, and the center-to-center distance is 6cm, the positional relationship between the two circles is () .

A. Inscribed B. Intersection C. Exo-D. Deviation 6. Known that {an} is an equi-ratio series, and an0, a2a4+2a3a5+a4a6=25, then the value of a3+a5 is equal to () The maximum value of the function f(x)=sinx-cosx is (), 28. The three rib lengths of the rectangle ABCD-A1B1C1D1 are AA=1, AB=2, and AD=4, starting from point A, along the cuboid The shortest distance from the surface to C is (), a number is rounded to an approximate value of 30,000, and the maximum value of this number is (), the image of the known inverse function y=k passes through the point p(-1, 2), then this function The image is located at ().

xA. Second and third quadrants B, first and third quadrants C. Third, fourth quadrant D, second and fourth quadrants 11, one bag containing five black balls, three white balls, and the other bag With 4 black balls and 4 white balls, one ball is taken from each of the two bags. The probability that both balls are black balls is (), the known vector a=(5, -3), then a=( There is a kind of food made up of 3 kilograms of milk sugar of 30 yuan per kilogram, 3 kilograms of milk of 6 yuan per kilogram, 4 kilograms of 15 yuan per kilogram of essence powder, and finally the average price per kilogram of this food is ( )yuan.

Known AUBM, AIBN, the following relationship is correct () == N15. The three digits that can be composed of four numbers 0, 1, 2, 3 without repeating numbers are () Fill in the blanks 1. The known curve f(x, y) = 0 satisfies f(-x, -y) = 0, then the curve is symmetric about _________.

The volunteers arranged for 6 people to participate in community charity activities on Saturday and Sunday.

If there are 3 people arranged each day, there are __________ different arrangements. 3. The slope of the tangent at (1,1) of the function y=2x3-x2+x-1 is __________. 4. Master Li randomly checked the daily water consumption (unit: ton) of the unit for 6 days in April this year. The results are as follows: 7,8,8,7,6,6. Based on these data, it is estimated that the water consumption in April is _ _________Ton. 5. A ball falls freely from a height of 100 meters, and each time it hits the ground, it jumps back to half of the original height. When it landed for the 10th time, it passed ____________ meters. 6. The monotonically increasing interval of the function y=2x+1 is ___________. X7. Known set M={X鈭?3x5}, N={x鈭?5x5}, then MN__________. 8. It is known that F is the left focus of the hyperbola, A(1,4), and P is the moving point on the right branch of the hyperbola. The minimum value of 鈭F鈭?鈭A鈭?is __________. 29. Knowing f(1-cosx)=sinx, then f(x)=_________.10, if: A=2脳2脳5, B=2脳3脳5, then the greatest common divisor of A and B is ____, the least common multiple is _____.11. If 0 is set, then sin is equal to __________. Sincos12. The distance from point p(1,2) to line y=2x+1 is __________.2213. If p(2,1) is the midpoint of AB of the circle of circle (x-1)+y=25, Then the equation of the straight line AB is _________. Second, the calculation question. 1. Known function f(x)=x-2x.(I) Find the monotonic interval of the function y=f(x) and indicate whether it is an increasing function or a decreasing function in each monotone interval; (II) Find the function y =f(x) The maximum and minimum values 鈥嬧€媜ver the interval [0,4]. 2. Construct a rectangular parallelepiped pool with a volume of 4,800 cubic meters and a depth of 3 meters. If the cost of the bottom and the wall of the pool is 150 yuan and 120 yuan per square meter respectively, then how to design the pool can make the total cost the lowest and lowest. What is the total cost? 23. Assume that the images of the two quadratic functions are symmetric about the line x=1, where the expression of one function is y=x+2x-1, and the expression of the other function is obtained. 4. When the original price of a book is a yuan per unit, the total amount sold is b. If the price increases by x%, the total amount of sales is expected to decrease by %, and the total sales amount of this book is asked when x is the value. maximum. 5, set the number of columns {an}, {bn} satisfies a1=1, b1=0 and an12an3bn, n=1,2,3...bn1an2bn6. The center of the known circle O is at the coordinate origin, the circle O and the x-axis positive half-axis Cross at point A, intersect with the positive semi-axis of y-axis at point B, -AB-=22. Let P be a point on circle O, and OP鈭B, find the coordinates of point P. Part 2: Primary school mathematics teacher recruitment professional knowledge mathematics teacher recruitment examination professional knowledge review one, review requirements (because the recruitment subject is only knowledge, so this content is the college entrance examination knowledge points) 1, understand the collection and representation, master subsets, Complete set and complement, subset and union definition; 2, master the solution with absolute value inequality and unitary quadratic inequality; 3, understand the meaning of logical conjunction, will skillfully transform four propositions, master the counter-evidence; Understanding the sufficient conditions, the necessary conditions and the meaning of the necessary and sufficient conditions will judge the necessary and sufficient relationship between the two propositions; 5. Learn to use the definition to solve the problem, understand the combination of number and shape, classification discussion and equivalent transformation. Second, study guidance 1, the concept of collection: (1) element characteristics in the set, certainty, mutuality, disorder; (2) classification of the collection: 1 by the number of elements: finite set, infinite set; 2 Element feature points; number sets, point sets. For example, the set {y|y=x2} represents a non-negative real set, and the point set {(x, y)|y=x2} represents a parabola with the opening upward and the y-axis as the axis of symmetry; (3) the representation of the set : 1 enumeration method: used to represent a finite set or an infinite set with significant laws, such as N + = {0, 1, 2, 3,}; 2 description method. 2, two types of relations: (1) the relationship between elements and collections, with or; (2) the relationship between the collection and the collection, with, = =, when AB, said A is a subset of B; when AB, Let A be the true subset of B. 3, set operation (1) intersection, and, complement, definition: A 鈭?B = {x | x 鈭?A and x 鈭?B}, A 鈭?B = {x | x 鈭?A, or x 鈭?B}, CUA = {x|x鈭圲, and xA}, the set U represents the complete set; (2) the law of operation, such as A鈭?B鈭狢)=(A鈭〣)鈭?A鈭〤), CU(A鈭〣)= (CUA) 鈭?(CUB), CU (A 鈭?B) = (CUA) 鈭?(CUB), and the like. 4, proposition: (1) proposition classification: true proposition and false proposition, simple proposition and compound proposition; (2) the form of compound proposition: p and q, p or q, non-p; (3) the true and false of the composite proposition: For p and q, when q and p are true, it is true; when one of p and q is false, it is false. For p or q, when p and q are both false, it is false; when one of p and q is true, it is true; when p is true, non-p is false; when p is false When non-p is true. (3) Four propositions: remember that if 'p is q' is the original proposition, then the proposition is 'if non-p is non-q', the inverse proposition is 'if q is p', and the inverse proposition is 'if non-q is non-p '. The two propositions in which each is counter-inverse is true and false, that is, equivalent. Therefore, the number of four propositions can only be an even number. 5. Sufficient conditions and necessary conditions (1) Definition: For the proposition 'if p is q', when it is a true proposition, p is a sufficient condition for q, and q is a necessary condition for p, when its inverse proposition is In true time, q is a sufficient condition for p, p is a necessary condition for q. When both propositions are true, p is said to be a necessary and sufficient condition for q; (2) when judging sufficient conditions and necessary conditions, it is first necessary to distinguish which The proposition is a condition, which proposition is a conclusion, and secondly, the conclusion is divided into four situations: sufficient unnecessary conditions, necessary insufficient conditions, sufficient and necessary conditions, neither sufficient nor necessary. From the set point of view, if all the objects satisfying the condition p constitute the set A, and all the objects satisfying the condition q constitute the set B, then when AB, p is a sufficient condition for q. In the case of BA, q is a sufficient condition for p. When A=B, p is a necessary and sufficient condition for q; (3) When p and q are mutually necessary and sufficient, the idea of 鈥嬧€媡he equivalence conversion of the proposition is embodied. 6. The counter-evidence method is an important method of middle school mathematics. Some algebraic propositions will be proved by the counter-evidence method. 7. The concept of collection and its basic theory are one of the most basic contents of modern mathematics.

Learn to use mathematical ideas to solve mathematical problems.

3. Typical example 1. The known set M={y|y=x2+1, x鈭圧}, N={y|y=x+1, x鈭圧}, find M鈭㎞.

analysis of problem-solving ideas: Before the set operation, we must first identify the set, that is, recognize the characteristics of the elements in the set.

M and N are both sets of numbers, which cannot be mistaken for point sets, thus solving the equations.

Secondly simplifies the collection, or makes the characteristics of the collection clear.

M={y|y=x2+1, x鈭圧}={y|y鈮?}, N={y|y=x+1, x鈭圧}={y|y鈭圧} 鈭碝鈭㎞=M={y|y鈮?} Description: In fact, from the function point of view, M and N in this question are the value fields of the quadratic function and the primary function, respectively.

In general, the set {y|y=f(x), x鈭圓} should be considered as the value range of the function y=f(x), which is obtained by finding the function range.

This set is essentially different from the set {(x,y)|y=x2+1,x鈭圧}, which is a set of points representing all points on the parabola y=x2+1, belonging to the graph category.

The element features in the collection are independent of the letters representing the elements, for example {y|y鈮?}={x|x鈮?}.

Example 2, known set A={x|x2-3x+2=0}, B+{x|x2-mx+2=0}, and A鈭〣=B, find the range of real numbers m.

Analysis of problem-solving ideas: A={1,2} is reduced, and A鈭〣=BBA is discussed according to the set of elements B in the set, B=蠁, B={1} or {2}, B={1,2} When B=蠁, 鈻?m2-80鈭?2m22 When B={1} or {2}, when B={1,2}, 鈭磎=3 , m=3 or 22m22 Description: Classification discussion is an important idea of 鈥嬧€媘iddle school mathematics. Comprehensively excavating the hidden condition in the title is an important aspect of the quality of problem solving. If this question is B={1} or {2}, it cannot be missed. =0.

Example 3, prove by the counter-evidence method: x, y鈭圧, x+y鈮? are known, and at least one of x and y is greater than 1.

Analysis of problem-solving ideas: assuming x1 and y1, the property x+y2 added by the inequality is inconsistent with the known x+y鈮?. The assumption is not true 鈭磝, y at least one is greater than 1; The theoretical basis is: to prove that 'if p is q' is true, the proficiency 'if p is not q' is false, because under condition p, q and non-q are opposite events (cannot be established at the same time, but there must be a ), so when 'if p is not q' is false, 'if p then q' must be true.

Example 4: If A is a necessary and insufficient condition for B, C is a necessary and sufficient condition for B, and D is a sufficient and unnecessary condition of C, and it is judged what condition D is A.

Analysis of problem-solving ideas: Using the '' and '' symbols to analyze the relationship between the propositions DCBA鈭碊A, D is the sufficient unnecessary condition of A: the symbols '', '' are transitive, but the former is single In the direction, the latter is bidirectional.

Example 5: Find a straight line: ax-y+b=0 The necessary and sufficient condition for the intersection of two straight lines 1:2x-2y-3=0 and 2:3x-5y+1=0.

Analysis of problem-solving ideas: starting from the necessity, proof of both sufficiency and necessity.

0, m has no solution 1m20 or 42m2012m122 from 2x2y301711, 1, 2 intersection P(,) 443x5y10鈭礟 point a鈭?711b044鈭?7a+4b=11 Sufficiency: Let a, b satisfy 17a+4b=11鈭碽1117a41117a04 Substituting the equation: axy finishing: (y1117)a(x)044111717110, the intersection of x0 (,) 4444 This equation shows that the straight line is over two straight lines y and this point is the intersection of 1 and 2. Said, the proposition is true: there are generally two ways to prove the necessary and sufficient conditions, one is to use '', two-way transmission, at the same time to prove the adequacy and necessity; the other is to prove the necessity and sufficientness respectively. Start with necessity and test the adequacy.

Fourth, synchronous practice (a) multiple choice questions 1, set M = {x | x2 + x + 2 = 0}, a = lg (lg10), then the relationship between {a} and M is A, {a} =MB, M{a}C, {a}MD, M{a}2, known corpus U=R, A={x|xa|2}, B={x|x-1|鈮?}, And A 鈭?B = 蠁, then the value range of a is A, [0, 2] B, (-2, 2) C, (0, 2] D, (0, 2) 3, known set M = {x|x=a2-3a+2, a鈭圧}, N={x|x=b2-b, b鈭圧}, then the relationship between M and N is A, MNB, MNC, M=ND, 鈥嬧€媙o Determine 4, set the set A = {x | x 鈭?Z and -10 鈮?x 鈮?-1}, B = {x | x 鈭?Z, and | x | 鈮?5}, then the number of elements in A 鈭?B is A, 11B, 10C, 16D, 155, set M = {1, 2, 3, 4, 5} subsets are A, 15B, 16C, 31D, 326, for the proposition 'the four inner angles of the square are equal', below The correct judgment is A, the given proposition is false B, its inverse no proposition is true C, its inverse proposition is true D, its no proposition is true 7, and '伪鈮犖? is cos伪鈮燾os尾' Fully unnecessary condition B, necessary insufficient condition C, necessary and sufficient condition D, neither sufficient nor necessary condition 8, set A={x|x=3k-2, k鈭圸}, B={y|y =3+1, 鈭圸}, S={y|y=6m+1, m鈭圸} is the relationship between A, SBAB S=BAC, SB=AD, SB=A9, and the necessary and sufficient conditions for the equation mx2+2x+1=0 to have at least one negative root are A, 0m鈮? or m0B, 0m鈮?C, m1D, m鈮?10, known p: the equation x2+ax+b=0 has and only has an integer solution, q: a, b is an integer, then p is a of q, sufficient unnecessary condition B, necessary insufficient condition, necessary condition D, neither sufficient It is also unnecessary to condition (2) fill in the blank question 11, known M = {m|m4x3Z}, N = {x|N}, then M鈭㎞ = ____ empty set ______.

2212. Among the 100 students, there are 60 table tennis fans and 65 volleyball fans. The minimum number of people who like them is ___25__ people.

Maximum __60_ people 13, 14, 15, and the equation for x |x|-|x-1|=a The necessary and sufficient condition for the solution is ________________.

The proposition 'If ab=0, then at least one of a and b is zero' is the inverse of the proposition _____ true proposition _______.

The non-empty set p satisfies the following two conditions: (1) p(2) If the element a鈭坧, then 6-a鈭坧, the set p number is {1, 2, 3, 4, 5} , ____7______.

(3) Answer questions 16, 17, 18, 19, function one, review requirements 7, definition of function and generality; 2, the use of function properties.

Second, learning guide 1, the concept of function: (1) mapping: set non-empty number set A, B, if any element a in set A, there is a unique element b in set B corresponding to it, then The correspondence from A to B is referred to as a map, denoted as f: A 鈫?B, f represents the corresponding law, and b = f (a).

If the images of different elements in A are different, the map is said to be a single shot. If each element in B has an original image corresponding to it, the map is said to be a full shot.

A mapping that is both single-shot and full-shot is called a one-to-one mapping.

(2) Function definition: A function is a map defined on a non-null set A, B. At this time, the scale set A is a domain, and the image set C={f(x)|x鈭圓} is a value. area.

Definition domain, corresponding law, value domain constitutes the three elements of the function. Logically, the domain is defined, and the corresponding law determines the value domain. It is the two most basic factors.

Inversely, the range also limits the domain.

Find the function definition field by solving the inequality (group) of the independent variables.

It is necessary to memorize the domain of the basic elementary function. The elementary function consists of four arithmetic operations whose domain is the intersection of each elementary function domain.

Composite function domain, not only to consider the domain of the inner function, but also to consider that the outer function is known to ax2, b=2-x, c=x2-x+1, and prove by a proof: a, b, c At least one of them is not less than one.

12Set the set A={(x,y)|y=ax+1}, B={(x,y)|y=|x|}, if A鈭〣 is a set of singletons, find a Range of values.

The known parabola C:y=-x2+mx-1, the point M(0,3), N(3,0), and the necessary and sufficient conditions for the parabola C and the line segment MN have two different intersections.

Let A={x|x2+px+q=0}鈮犗? M={1,3,5,7,9}, N={1,4,7,10}, if A鈭㎝ =蠁, A鈭㎞=A, find the values 鈥嬧€媜f p and q.

Corresponding to the requirements of the law.

Understanding the function definition domain should be closely related to the corresponding rules.

Functional domain is the basis and premise for studying the nature of functions.

The function correspondence rules usually appear as tables, analytic and images.

The analytical form is the most common form of expression.

The method of finding the analytic formula of the known type function is the undetermined coefficient method, the analytic formula of the abstract function is commonly used to change the element method and the merging method.

Functional value range is a common problem in functions. In the scope of elementary mathematics, the direct method has monotonicity, basic inequality and geometric meaning. The indirect method is the function of function and equation, which is expressed as 鈻?method, anti Function method, etc., in the higher mathematics range, it is more convenient to use the derivative method to find the maximum value (extreme value) of some functions.

There is a typical problem of finding a range of values 鈥嬧€媔n all parts of middle school mathematics. A typical treatment method is to establish a function analytic method by means of a function value field.

2, the generality of the function (1) Parity: The symmetry of the function domain with respect to the origin is a necessary condition for judging the parity of the function. When using the definition judgment, it should be performed after the simplification of the analytic expression, and the domain is flexibly applied. The deformation, such as f(x)f(x)0, the geometric meaning of parity is two special image symmetry.

The parity of a function is a universal property on a domain, and a definition is an identity on a domain.

The step of determining the parity can be simplified by the arithmetic nature of the parity.

(2) Monotonicity: The monotonicity of the study function should be combined with the monotonic interval of the function, and the monotonic interval should be a subset of the domain.

Methods for judging the monotonicity of a function: 1 definition method, that is, ratio difference method; 2 image method; 3 monotonic operation property (substantially inequality property); 4 compound function monotonicity judgment rule.

The monotonicity of a function is a generally established property on a monotonic interval, and is an inequality that is constant over a monotonic interval.

The monotonicity of a function is the most active property of a function. Its application is mainly reflected in inequalities, such as comparison size, solution abstract function inequality, and so on.

(3) Periodicity: Periodicity is mainly used in trigonometric functions and abstract functions, and is an important means of returning to the mind.

Important methods for finding cycles: 1 definition method; 2 formula method; 3 image method; 4 use important conclusion: if function f(x) satisfies f(ax)=f(a+x), f(bx) =f(b+x), a鈮燽, then T=2|ab|.

(4) Inverse function: Whether the function is an inverse function is one of the important applications of the function concept. Before the inverse function, first determine whether the function has an inverse function, and the inverse function f-1 of the function f(x) The nature of x) is closely related to the nature of f(x), such as the definition domain, the value domain interchange, the same monotonicity, etc., and the problem of the inverse function f-1(x) is classified as the function f(x). It is an important idea to deal with the inverse function problem.

Let the function f(x) be defined as A, and the value field be C, then f-1[f(x)]=x, x鈭圓f[f-1(x)]=x, x鈭圕3, The image of the function function of the function is not only an important aspect of the function property, but also can intuitively reflect the nature of the function. In the process of solving the problem, the image tool is fully utilized.

Image method: 1 plot method; 2 image transform.

You should master common image transformations.

4, the common elementary function of this order; primary function, quadratic function, inverse proportional function, exponential function, logarithmic function.

Understand the generality of functions under specific correspondence rules, and grasp the nature of these specific rules.

The piecewise function is an important function model.

For abstract functions, it is usually to grasp the function property is the identities on the domain, and use the assignment method (variable substitution method) to solve the problem.

Contacting a specific function model can easily find solutions to problems and solve problems.

The application question is an important question type for the use of functions.

Check the meaning of the question, find the quantitative relationship, and grasp the model is the key to solving the application problem.

5, the main method of thinking: number combination, classification discussion, function equations, reversion and so on.

Three, typical example example 1, known f(x) analysis: 2x3, the image of the function y=g(x) and the image of y=f-1(x+1) are symmetric about the straight line y=x, Find the value of g(11).

x1f(x)1(f(x)鈮?).

f(x) Part 3: Mathematics Teacher Recruitment Examination Professional Knowledge Mathematics Teacher Recruitment Examination Professional Knowledge Review 1. Review Requirements 1. Understanding Sets and Representations, Master Subsets, Complete Sets and Complements, Subsets and Unions Definition; 2. Master the solution with absolute value inequality and unitary quadratic inequality; 3. Understand the meaning of logical connection words, skillfully transform four propositions, master the counter-evidence method; 4. Understand sufficient conditions, necessary conditions and necessary and sufficient conditions Meaning, will judge the necessary and sufficient relationship between the two propositions; 5, learn to solve problems with definitions, understand the combination of number and shape, classification discussion and equivalent transformation and other ideas.

There is p', the inverse proposition is 'if it is not q, it is not p'.

The two propositions in which each other is reversed are true and false, that is, equivalent.

Therefore, the number of four propositions can only be an even number.

5, Sufficient conditions and necessary conditions (1) Definition: For the proposition 'if p then q', when it is a true proposition, p is a sufficient condition for q, q is a necessary condition for p, when it is When the inverse proposition is true, q is a sufficient condition for p, and p is a necessary condition for q. When both propositions are true, p is a necessary and sufficient condition for q; (2) when judging sufficient conditions and necessary conditions, first It is necessary to distinguish which proposition is a condition and which proposition is a conclusion. Secondly, the conclusion must be divided into four situations: sufficient unnecessary conditions, necessary insufficient conditions, sufficient and necessary conditions, neither sufficient nor necessary.

From the perspective of the set, if all the objects satisfying the condition p constitute the set A, and all the objects satisfying the condition q constitute the set q, then when AB, p is a sufficient condition for q.

When

BA, p is a sufficient condition for q.

A=B, p is a necessary and sufficient condition for q; (3) When p and q are mutually necessary and sufficient, the idea of 鈥嬧€媡he equivalence transformation of the proposition is embodied.

Second, learning guidance 1, the concept of collection: (1) element characteristics in the set, certainty, mutuality, disorder; (2) classification of the set: 1 by number of elements: finite set, infinite set ; 2 according to element characteristics; number set, point set.

As the set of numbers {y|y=x}, represents a set of non-negative real numbers, the set of points {(x, y)|y=x} represents a parabola with the opening upward, with the y-axis as the axis of symmetry; (3) set Representation: 1 enumeration method: used to represent a finite set or an infinite set with significant laws, such as N+={0,1,2,3,}; 2 description method.

2, two types of relationship: (1) the relationship between elements and collections, with or; (2) the relationship between the collection and the collection, with, = =, when AB, A is called a subset of B; At AB, the 226 and the counter-evidence method are important methods for middle school mathematics.

Prove some algebraic propositions with a counter-evidence method.

7. The concept of collection and its basic theory are one of the most basic contents of modern mathematics.

Learn to use mathematical ideas to solve mathematical problems.

Third, the typical example of the example 1, the known set M = {y | y = x + 1, x 鈭?R}, N = {y | y = x + 1, x 鈭?R}, find M 鈭?N.

analysis of problem-solving ideas: Before the set operation, we must first identify the set, that is, recognize the characteristics of the elements in the set.

M and N are both sets of numbers, which cannot be mistaken for point sets, thus solving the equations.

Secondly simplifies the collection, or makes the characteristics of the collection clear.

M={y|y=x+1, x鈭圧}={y|y鈮?}, N={y|y=x+1, x鈭圧}={y|y鈭圧} 鈭碝鈭㎞=M={y|y鈮?} Description: In fact, from the function point of view, M and N in this question are the value fields of the quadratic function and the primary function, respectively. In general, the set {y|y=f(x), x鈭圓} should be regarded as the value range of the function y=f(x), which is obtained by finding the function range. This set is essentially different from the set {(x,y)|y=x+1,x鈭圧}, which is a set of points, representing all points on the parabola y=x+1, belonging to the graphical category. The element features in the set are independent of the letters that represent the elements, for example {y|y鈮?}={x|x鈮?}. Example 2. The known set A = {x | x - 3x + 2 = 0}, B + {x | x - mx + 2 = 0}, and A 鈭?B = B, find the real range of m. Analysis of problem-solving ideas: Simplification conditions are A={1,2}, A鈭〣=BBA is discussed according to the set of elements in the set B, B=蠁, B={1} or {2}, B={ 1,2} When B=蠁, 螖=m-80鈭?2m222222222A is the true subset of B. 3, set operation (1) intersection, and, complement, definition: A 鈭?B = {x | x 鈭?A and x 鈭?B}, A 鈭?B = {x | x 鈭?A, or x 鈭?B}, CUA = {x|x鈭圲, and xA}, the set U represents the complete set; (2) the law of operation, such as A鈭?B鈭狢)=(A鈭〣)鈭?A鈭〤), CU(A鈭〣)= (CUA) 鈭?(CUB), CU (A 鈭?B) = (CUA) 鈭?(CUB), and the like. 4, proposition: (1) proposition classification: true proposition and false proposition, simple proposition and compound proposition; (2) the form of compound proposition: p and q, p or q, non-p; (3) the true and false of the composite proposition: For p and q, when q and p are true, it is true; when one of p and q is false, it is false. For p or q, when p and q are both false, it is false; when one of p and q is true, it is true; when p is true, non-p is false; when p is false When non-p is true. (3) Four propositions: remember that if 鈥渜 then p鈥?is the original proposition, then the proposition is 鈥渋f non-p is non-q鈥? and the inverse proposition is 鈥渋f q0 is B={1} or {2}, m No solution 1m20 or 42m2012m When B={1,2}, 122 sufficiency: Let a, b satisfy 17a+4b=11鈭碽1117a4 Substitute the equation: axy finishing: (y鈭磎=3 in summary, m =3 or 22m2 Description: Classification discussion is an important idea of 鈥嬧€媘iddle school mathematics. Comprehensively excavating the hidden condition in the problem is an important aspect of the problem solving quality. For example, when B={1} or {2}, the 鈻?0 can not be missed. Example 3. Prove that the x, y 鈭?R, x + y 鈮?2 are known by the counter-evidence method, and at least one of x and y is greater than 1. The solution to the problem is to assume that x1 and y1 are added by the inequality in the same direction. The property x+y2 is inconsistent with the known x+y鈮?. The assumption is not true. At least one of yx and y is greater than 1; the theoretical basis of the counter-evidence method is: 鈥淚f p is q鈥?is true, the proficiency 鈥渋f p Then non-q' is false, because under condition p, q and non-q are opposite events (cannot be established at the same time, but one must be established), so when 'if p is not q' is false, 'if p is q' 'It must be true. Example 4, if A is necessary for B, it is not sufficient. C, is the necessary and sufficient condition of B, D is the sufficient and unnecessary condition of C, and judges what condition D is A. Analysis of problem solving: using '', '' symbol to analyze the relationship between each proposition DCBA鈭碊A , D is a sufficient unnecessary condition of A: the symbols '', '' have transitivity, but the former is unidirectional, the latter is bidirectional. Example 5, seeking a straight line: ax-y + b = 0 after two The necessary and sufficient conditions for the intersection of straight line 1:2x-2y-3=0 and 2:3x-5y+1=0. Analysis of problem-solving ideas: starting from necessity, proof of sufficiency and necessity. 2x2y301711 , 2 intersection point P (,) 443x5y101117a041117) a (x) 044111717110, the intersection of x0 (,) 4444 This equation shows that the line is always over two straight lines y and this point is the intersection of 1 and 2 鈭?sufficiency As stated, the proposition is true: there are generally two ways to prove the necessary and sufficient conditions. One is to use '', two-way transmission, at the same time to prove the adequacy and necessity; the other is to prove the necessity and sufficientness respectively. Start with the necessity, and then test the sufficiency. Fourth, synchronous practice (1) multiple choice questions 1, set M = {x | x + x + 2 = 0}, a = lg (lg10) , then the relationship between {a} and M is A, {a}=MB, M{a}C, {a}MD, M{a}2, known full set U=R, A={x|xa|2 }, B={x|x-1|鈮?}, and A鈭〣=蠁, then the range of a is A, [0, 2] B, (-2, 2) C, (0, 2 ]D, (0,2)3, known set M={x|x=a-3a+2, a鈭圧}, N, {x|x=bb, b鈭圧}, then M, N The relationship is A, MNB, MNC, M=ND, 鈥嬧€媢ncertainty 4, set A={x|x鈭圸 and -10鈮鈮?1}, B={x|x鈭圸, and |x|鈮?5}, then the number of elements in A鈭狟 is A, 11B, 10C, 16D, 155, and the subset of sets M={1, 2, 3, 4, 5} is A, 15B, 16C, 31D, 326 For the proposition 'the four inner angles of the square are equal', the following is correct A, the given proposition is false B, its inverse no proposition is true C, its inverse proposition is true D, and its no proposition is true 7 '伪鈮犖? is A of cos伪鈮燾os尾鈥? sufficient unnecessary condition B, necessary insufficient condition C, necessary and sufficient condition D, neither sufficient nor necessary condition 8, set A={x|x=3k- 2, k鈭圸}, B={y|y=3+1, 鈭圸}, S={y|y=6m+1, m鈭圸} is the relationship between 222鈭佃繃鐐?P鈭碼1711b044鈭?17a+4b=11A, SBAB, S=BAC, SB=AD, SB=A function I. 9. The necessary and sufficient condition for the equation mx+2x+1=0 to have at least one negative root is A, 0m鈮? or m0B, 0m鈮?C, m1D, m鈮?10, known p: equation x+ax+b=0 There are only integer solutions, q: a, b is an integer, then p is the A of q, sufficient unnecessary condition B, necessary insufficient conditions, necessary and sufficient conditions D, neither sufficient nor necessary conditions (2) fill in the blank question 11 Known M={m|22 II. Learning Guide 1. The concept of function: (1) Mapping: Set non-empty number set A, B. If there is any element a in set A, there is a unique element in set B. b corresponds to this, then the correspondence from A to B is the mapping, denoted as f: A 鈫?B, f represents the corresponding law, b = f (a). If the images of different elements in A are different, the mapping is said to be a single shot. If each element in B has an original image corresponding to it, the mapping is said to be a full shot. A mapping that is both single shot and full shot is called a one-to-one mapping. (2) Function definition: The function is the mapping defined on the non-null set A, B. At this time, the set A is the domain, and the set C={f(x)|x鈭圓} is the value range. The definition domain, the corresponding rule, the value domain constitutes the three elements of the function. Logically speaking, the domain is defined, and the corresponding law determines the value domain. It is the two most basic factors. In turn, the range also limits the domain. Find the function definition field by solving the inequality (group) about the independent variable. To memorize the domain of the basic elementary function, the elementary function consists of four arithmetic operations whose domain is the intersection of each elementary function domain. The composite function definition domain must consider not only the domain of the inner function but also the requirements of the corresponding law of the outer function. To understand the function definition domain, you should closely follow the corresponding rules. The function definition domain is the basis and premise for studying the nature of the function. The function correspondence rules usually appear as tables, analytic and images. The analytical form is the most common form of expression. The method for finding the analytic expression of the known type function is the undetermined coefficient method, and the analytic formula of the abstract function is commonly used for the meta-method and the merging method. Finding the function range is a common problem in functions. In the primary mathematics range, the direct method has monotonicity, basic inequality and geometric meaning. The indirect method is the function and equation, which is represented by 鈻?method, inverse function method, etc. In the higher mathematical range m4x3Z}, N={x|N}, then M鈭㎞=__________. 2212. Among the 100 students, there are 60 table tennis enthusiasts and 65 volleyball enthusiasts. The minimum number of people who love each other is ________. 13. The necessary and sufficient condition for the equation x to be x|-|x-1|=a is ________________. 14. The proposition 'If ab = 0, then at least one of a and b is zero' is the inverse of the proposition ____________. 15. The non-empty set p satisfies the following two conditions: (1) p(2) if the element a鈭坧, then 6-a{1, 2, 3, 4, 5}, 鈭坧, then the set p number is __________. (3) Solving the question 16, set the set A={(x,y)|y=ax+1}, B={(x,y)|y=|x|}, if A鈭〣 is a single element set, Find a range of values. 17. Knowing the parabola C: y=-x+mx-1, the point M(0,3), N(3,0), and the necessary and sufficient conditions for the parabola C and the line segment MN have two different intersections. 18. Let A={x|x+px+q=0}鈮犗? M={1,3,5,7,9}, N={1,4,7,10}, if A鈭㎝= 蠁, A鈭㎞=A, find the values 鈥嬧€媜f p and q. 19. Known in ax222, it is more convenient to use the derivative method to find the maximum value (extreme value) of some functions. In the various parts of middle school mathematics, there is a typical problem of finding a range of values. A typical processing method is to establish a function analytic method by means of a function value field. 2. The generality of the function (1) Parity: The symmetry of the function definition domain with respect to the origin is a necessary condition for judging the parity of the function. When using the definition judgment, it should be performed after the analytic expression is simplified, and the deformation of the domain is flexibly applied. For example, f(x)f(x)0, 鈮?). f(x)1(f(x)f(x)12, b=2-x,c=x-x+1, proved by the proof: at least one of a, b, c is not less than 1. 2 parity The geometric meaning is two special image symmetry. The parity of a function is a universal property on the domain, and the definition is an identity on the domain. The performance of the parity can simplify the step of determining the parity. (2) Monotonicity: The monotonicity of the study function should be combined with the monotonic interval of the function, and the monotonic interval should be a subset of the domain. The method of determining the monotonicity of the function: 1 definition method, ie the ratio difference method; 2 image method; 3 monotonic operation Nature (essentially inequality); 4 monotonic judgment rule of compound function. Function monotonicity is a general property of monotonic interval, which is a constant inequality on monotone interval. Function monotonicity is the most active property of function. Its application is mainly reflected in inequalities, such as comparing size, solving abstract function inequalities, etc. (3) Periodicity: Periodicity is mainly used in trigonometric functions and abstract functions, which is an important means of regressive thinking. :1 definition Method; 2 formula method; 3 image method; 4 use important conclusion: if the function f(x) satisfies f(ax)=f(a+x), f(bx)=f(b+x), a鈮燽 , T=2|ab|. (4) Inverse function: Whether the function is an inverse function is one of the important applications of the function concept. Before negating the function, first determine whether the function has an inverse function, the function f(x) The property of the inverse function f(x) is closely related to the property of f(x). For example, the domain, the value range are interchanged, have the same monotonicity, etc., and the problem of the inverse function f(x) is classified as the function f(x). The problem is to deal with the important idea of 鈥嬧€媡he inverse function problem. Let the function f(x) be defined as A and the value field be C, then f[f(x)]=x, x鈭圓f[f(x)]=x, The image of the image function of x鈭圕8 and function is not only an important aspect of the function property, but also can directly reflect the nature of the function. In the process of solving the problem, the image tool can be fully utilized. Image method: 1 trace method ; 2 image transformation. Should master the common image transformation. 4, the common elementary function of this order; primary function, quadratic function, inverse proportional function, exponential function, logarithmic function. Under the specific corresponding law to understand the function of the pass Sexuality The nature of these specific correspondence rules. The piecewise function is an important function model. For abstract functions, it is usually to grasp the function property is the identities on the domain, and use the assignment method (variable substitution method) to solve the problem. Link to the specific function. The model can easily find the solution to the problem and solve the problem. The application question is the important question type of the function of the function. Clearing the meaning of the problem, finding the quantitative relationship, grasping the model is the key to solving the application problem. : combination of number and shape, classification discussion, function equation, transformation, etc. 1-1-1-1 uses the relationship of number form, it can be seen that y=g(x) is the inverse function of y=f(x+1).浠庤€屽寲g(x)闂涓哄凡鐭(x)銆傗埖y=f(x+1)鈭磝+1=f(y)鈭磝=f(y)-1鈭磞=f(x+1 )鐨勫弽鍑芥暟涓簓=f(x)-1鍗砱(x)=f(x)-1鈭磄(11)=f(11)-1=-1-1-132璇勬敞锛氬嚱鏁颁笌鍙嶅嚱鏁扮殑鍏崇郴鏄簰涓洪€嗚繍绠楃殑鍏崇郴锛屽綋f(x)瀛樺湪鍙嶅嚱鏁版椂锛岃嫢b=f(a)锛屽垯a=f(b)銆備緥2銆佽f(x)鏄畾涔夊湪锛?鈭烇紝+鈭烇級涓婄殑鍑芥暟锛屽涓€鍒噚鈭圧鍧囨湁f(x)+f(x+2)=0锛屽綋-1x鈮?鏃讹紝f(x)=2x-1锛屾眰褰?x鈮?鏃讹紝鍑芥暟f(x)鐨勮В鏋愬紡銆傝В棰樻€濊矾鍒嗘瀽锛氬埄鐢ㄥ寲褰掓€濇兂瑙i鈭礷(x)+f(x+2)=0鈭磃(x)=-f(x+2)鈭佃寮忓涓€鍒噚鈭圧鎴愮珛鈭翠互x-2浠寰楋細f(x-2)=-f[(x-2)+2]=-f(x)褰?x鈮?鏃讹紝-1x-2鈮?鈭磃(x-2)=2(x-2)-1=2x-5鈭磃(x)=-f(x-2)=-2x+5鈭磃(x)=-2x+5锛?x鈮?锛夎瘎娉細鍦ㄥ寲褰掕繃绋嬩腑锛屼竴鏂归潰瑕佽浆鍖栬嚜鍙橀噺鍒板凡鐭ヨВ鏋愬紡鐨勫畾涔夊煙锛屽彟涓€鏂归潰瑕佷繚鎸佸搴旂殑鍑芥暟鍊兼湁涓€瀹氬叧绯汇€傚湪鍖栧綊杩囩▼涓繕浣撶幇浜嗘暣浣撴€濇兂銆備緥3銆佸凡鐭(x)=-x-3锛宖(x)鏄簩娆″嚱鏁帮紝褰搙鈭圼-1锛?]鏃讹紝f(x)鐨勬渶灏忓€硷紝涓攆(x)+g(x)涓哄鍑芥暟锛屾眰f(x)瑙f瀽寮忋€傚垎鏋愶細鐢ㄥ緟瀹氱郴鏁版硶姹俧(x)瑙f瀽寮忚f(x)=ax+bx+c锛坅鈮?锛?2-1涓夈€佸吀鍨嬩緥棰?x3-1渚?銆佸凡鐭(x)锛屽嚱鏁皔=g(x)鍥捐薄涓巠=f(x+1)鐨勫浘璞″叧浜庣洿绾縴=x瀵圭О锛屾眰g(11)x1鐨勫€笺€傚垎鏋愶細鍒檉(x)+g(x)=(a-1)x+bx+c-3a10鐢卞凡鐭(x)+g(x)涓哄鍑芥暟c30a1鈭碿32f(a+b)=f(a)f(b)锛岋紙1锛夋眰璇侊細f(0)=1锛涳紙2锛夋眰璇侊細瀵逛换鎰忕殑x鈭圧锛屾亽鏈塮(x)0锛涳紙3锛夎瘉鏄庯細f(x)鏄疪涓婄殑澧炲嚱鏁帮紱锛?锛夎嫢f(x)路f(2x-x)1锛屾眰x鐨勫彇鍊艰寖鍥淬€傚垎鏋愶細锛?锛変护a=b=0锛屽垯f(0)=[f(0)]鈭礷(0)鈮?鈭磃(0)=1锛?锛変护a=x锛宐=-x鍒檉(0)=f(x)f(-x)鈭磃(x)1f(x)22鈭磃(x)=x+bx+3涓嬮潰閫氳繃纭畾f(x)鍦╗-1锛?]涓婁綍鏃跺彇鏈€灏忓€兼潵纭畾b锛屽垎绫昏璁恒€?b2b2bf(x)(x)3锛屽绉拌酱x2422锛?锛夊綋b鈮?锛宐鈮?4鏃讹紝f(x)鍦╗-1锛?]涓婁负鍑忓嚱鏁?鈭?f(x))minf(2)2b7鈭?b+7=1鈭碽=3锛堣垗锛夛紙2锛夊綋鐢卞凡鐭0鏃讹紝f(x)10褰搙0鏃讹紝-x0锛宖(-x)0鈭磃(x)10f(x)b锛?4b2鏃讹紙-1锛?锛?bb2f()324(f(x))min2鍙坸=0鏃讹紝f(0)=10鈭村浠绘剰x鈭圧锛宖(x)0锛?锛変换鍙杧2x1锛屽垯f(x2)0锛宖(x1)0锛寈2-x10鈭磃(x2)f(x2)f(x1)f(x2x1)1f(x1)鈭碽314鈭碽22锛堣垗璐燂級锛?锛夊綋b鈮?1锛宐鈮?鏃讹紝f(x)鍦╗-1锛?]涓婁负澧炲嚱鏁?鈭?f(x)min=f(1)=4-b鈭?-b=1鈭碽=3鈭磃(x)x22x3锛屾垨f(x)x33x3璇勬敞锛氫簩娆″嚱鏁板湪闂尯闂翠笂鐨勬渶鍊奸€氬父瀵瑰绉拌酱涓庡尯闂寸殑浣嶇疆鍏崇郴杩涜璁ㄨ锛屾槸姹傚€煎煙鐨勫熀鏈鍨嬩箣涓€銆傚湪宸茬煡鏈€鍊肩粨鏋滅殑鏉′欢涓嬶紝浠嶉渶璁ㄨ浣曟椂鍙栧緱鏈€灏忓€笺€備緥4銆佸畾涔夊湪R涓婄殑鍑芥暟y=f(x)锛宖(0)鈮?锛屽綋x0鏃讹紝f(x)1锛屼笖瀵逛换鎰忕殑a銆乥鈭圧锛屾湁鈭磃(x2)f(x1)鈭磃(x)鍦≧涓婃槸澧炲嚱鏁帮紙4锛塮(x)路f(2x-x)=f[x+(2x-x)]=f(-x+3x)鍙?=f(0)锛宖(x)鍦≧涓婇€掑鈭寸敱f(3x-x)f(0)寰楋細3x-x0鈭?x3璇勬敞锛氭牴鎹甪(a+b)=f(a)路f(b) 鎭掔瓑寮忕殑鐗圭偣锛屽a銆乥閫傚綋璧嬪€笺€傚埄鐢ㄥ崟璋冩€х殑鎬ц川鍘绘帀绗﹀彿鈥渇鈥濆緱鍒板叧浜巟鐨勪唬鏁颁笉绛夊紡锛屾槸澶勭悊鎶借薄鍑芥暟涓嶇瓑寮忕殑鍏稿瀷鏂规硶銆?2222銆娿€嬪嚭鑷細閾炬帴鍦板潃锛歨ttp:///news/杞浇璇蜂繚鐣?璋㈣阿!鏌ョ湅鏇村鐩稿叧鍐呭>>銆?/p>

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