Set is the basic knowledge of high school mathematics and one of the compulsory contents over the years. This paper mainly investigates the understanding and understanding of the basic concepts of set.

Example: set A={(x, y)|x2+mx-y+2=0}, B={(x, y)|x-y+ 1=0, and 0≤x≤2}, if A∩B≦, then the fact is that.

2 Determination of sufficient and necessary conditions

Sufficient condition, necessary condition and necessary and sufficient condition are important mathematical concepts, which are mainly used to distinguish the relationship between condition P and conclusion Q of a proposition.

Example: It is known that the quadratic equation with real coefficient x2+ax+b=0 of X has two real roots α and β. It is proved that | α | < 2 and | β|.

3 using vector method to solve the problem

The content of this section is mainly to help candidates use vector method to analyze and solve some related problems.

Example: in the triangle ABC, A(5,-1), b (- 1, 7), c (1, 2), find the center line of (1)BC.

The length of AM; (2) The length of the cab bisector AD; (3) The value of cost ABC.

4 three "quadratic" and their relationship

The three "quadratic functions" of a quadratic function, a quadratic equation and a quadratic inequality are important contents of middle school mathematics, which are rich in connotation and closely related, and are also tools for learning many contents including quadratic curves. Nearly half of the college entrance examination questions are related to these three "secondary" questions.

Example: It is known that for all real values of x, the value of quadratic function f(x)=x2-4ax+2a+ 12(a∈R) is nonnegative. Find the value range of the root of the equation about X =|a- 1|+2.

5 solving the resolution function

Solving the resolution function is one of the key contents of the college entrance examination, which needs attention.

Example: Given f(2-cosx)=cos2x+cosx, find f(x- 1).

Example: (1) It is known that the function f(x) satisfies f(logax)= (where a >；; 0, a≠ 1, x>0), find the expression of f(x).

(2) It is known that the quadratic function f(x)=ax2+bx+c satisfies | f (1) | = | f (-1) | = | f (0) | =1,and the expression of f(x) is found.

Function range of 6 and its solution

The range of function and its solution are one of the key contents of college entrance examination in recent years.

Example: let m be a real number, let M = {m | m> 1}, and f(x)=log3(x2-4mx+4m2+m+).

(1) proves that f(x) is meaningful to all real numbers when m∈M; On the other hand, if f(x) is meaningful to all real numbers x, then m ∈ m

(2) When m∈M, find the minimum value of function f(x).

(3) Verification: For each m∈M, the minimum value of function f(x) is not less than 1.

7 parity and monotonicity (1)

Monotonicity and parity of functions are one of the key contents of college entrance examination. It is necessary to master the judgment method and correctly understand the images of monotone function and parity function.

For example, let a>0, an even function on f(x)= r, (1) find the value of a; (2) It is proved that f(x) is a increasing function on (0, +∞).

8 parity and monotonicity (2)

Monotonicity and parity of functions are one of the key and hot topics in college entrance examination, especially the application of these two properties is more prominent. This section mainly helps candidates learn how to solve problems by using two properties, master basic methods and form application consciousness.

Example: It is known that even function f(x) is increasing function on (0, +∞), f(2)=0, and the inequality f[log2(x2+5x+4)]≥0 can be solved.

Example: it is known that odd function f(x) is a decreasing function defined on (-3,3) and satisfies the inequality f (x-3)+f (x2-3) < 0. Let the inequality solution set be a, B = A ∨{ x | 1≤x ≤} and find the function g (x).

9 exponential function, logarithmic function problem

Exponential function and logarithmic function are one of the key contents of college entrance examination.

Example: let f(x)=log2 and F(x)= +f(x).

(1) Try to judge the monotonicity of function f(x), define it by monotonicity of function, and give proof;

(2) If the inverse function of f(x) is f- 1(x), it is proved that for any natural number n(n≥3), there exists f-1(n) >:

(3) If the inverse function F- 1(x) of F(x), it is proved that the equation F- 1(x)=0 has a unique solution.

10 function image and image transformation

The image and nature of function is one of the key contents of college entrance examination. To master the general law of function image change, we can use the image of function to study the properties of function.

Example: Given the image with function f(x)=ax3+bx2+cx+d, find the range of B.

Synthesis of 1 1 function

The problem of function synthesis is one of the hot spots and key contents of college entrance examination over the years, and it is generally difficult.

Example: let the domain of function f(x) be R. For any real numbers x and y, there is f(x+y)=f(x)+f(y), when x >；; 0 when f (x)

(1) verification: f(x) is odd function;

(2) Find the maximum value of f(x) in the interval [-9,9].

Images and properties of 12 trigonometric function

The image and nature of trigonometric function is a hot spot in the college entrance examination. When reviewing, we should make full use of the idea of combining numbers with shapes, and combine image with nature. This section mainly helps candidates to master images and attributes and use them flexibly.

Example: it is known that α and β are acute angles, and x (α+β-) >; 0, try to prove the inequality f (x) = x

Example: let z 1=m+(2-m2)i, z2=cosθ+(λ+sinθ)i, where m, λ, θ∈R, and z 1=2z2, find the range of λ.

163 simplification and evaluation of formulas of trigonometric functions

Simplification and evaluation of trigonometric function is one of the key contents of college entrance examination. Through the study of this section, candidates can master the rules and methods of solving simplification and evaluation questions, especially some conventional simplification and evaluation skills, so as to optimize our problem-solving effect and get twice the result with half the effort.

Example: Known

Trigonometric function in 14 triangle

The trigonometric function relationship in triangle is one of the key contents of college entrance examination over the years.

● It is known that the three internal angles A, B and C of △ABC satisfy A+C=2B. Find the value of cos.

Proof strategy of inequality 15

The proof methods of inequality are flexible and diverse, and can be combined with various contents. The content of inequality proof and the proof of pure inequality have always been difficult points in high school mathematics. This difficulty focuses on cultivating students' mathematical deformation ability, logical thinking ability and ability to analyze and solve problems.

16 Solving Inequalities

Inequality is widely used in production practice and related disciplines, and it is also an important tool for learning advanced mathematics. Therefore, inequality is the focus of NMET mathematical proposition, and it is widely used to solve inequality, such as finding the definition and value range of function. In NMET, the solution of inequality is very demanding, which is often closely related to the concept and properties of functions, especially quadratic functions, exponential functions and logarithmic functions. Judging from the college entrance examination questions over the years, there are contents about solving inequalities every year, some of which are directly examined and some are indirectly examined.

Comprehensive application of inequality 17

Inequality is another important content after function and equation. As a tool to solve problems, its comprehensive application with other knowledge is more prominent. The application of inequality can be roughly divided into two categories: one is to establish inequality to find the range of parameters or solve some practical application problems; The other is to establish a functional relationship and solve the maximum problem by using mean inequality. This difficulty provides relevant thinking methods, so that candidates can use the properties, theorems and methods of inequality to solve problems in functions, equations, practical applications and so on.

For example, let the quadratic function f (x) = AX2+BX+C (A > 0), two roots of the equation f(x)-x=0, x 1, x2 satisfies 0.

(1) When x∈[0, x 1, prove x.

(2) Let the image of the function f(x) be symmetrical about the straight line x=x0, and prove that x0 < x0.