Senior one is a critical period for mathematics learning. I found that many excellent students in primary and junior high schools stumbled in mathematics when they entered the high school stage. To learn high school mathematics well, we need to have an overall understanding and grasp of high school mathematics knowledge.

gather

When entering high school, the first lesson in mathematics is assembly. Abstract concepts and symbolic terms are the salient features of set units, such as the concepts of intersection, union and complement and their representations, the relationship between sets and elements and their representations, the relationship between sets and representations, the definitions of subsets, proper subset and sets, and so on. The elements in the set have "three properties": (1) certainty: the elements in the set should be certain and cannot be ambiguous. (2) Reciprocity: The elements in the set should be different from each other, and the same element can only be counted as one in the set. (3) Disorder: The elements in the set are out of order.

For example: known set m = {x | x? 0? 5+x-6 = 0} Let n = {y | ay+2, a ∈ r}, and n ∩ cum = φ, real number a =?

Solution: Because n ∩ cum = φ, so n? 6? 7 meters

Because m = {x | x? 0? 5+x-6 = 0} = {-3,2} so n = {2} or {-3} or {-3,2}

When n = φ, a = 0.

When n = {2}, 2a+2 = 0 and a =- 1.

When n = {-3}, -3a+2 = 0 and a = 2/3.

So the real number A = 0 or A =- 1 or A = 2/3.

Note: Don't forget the situation when φ.

inequality

As for the absolute value of (1), we can consider removing the absolute value. The methods to remove the absolute value include: discussing that the part within the absolute value is greater than, equal to and less than zero; Square the two sides to remove the absolute value; It should be noted that both sides of the inequality sign are non-negative. Inequalities with multiple absolute value symbols can be solved by the method of "discussing by zero points"

(2) The solution of fractional inequality: the general solution is transformed into algebraic expression inequality; (3) Solution of inequality group: Find the solution set of each inequality in the inequality group, and then find its intersection, which is the solution set of this inequality group. In the intersection, the solution set of each inequality is usually drawn on the same number axis, and their common parts are taken. (4) Solving inequalities with parameters: When solving inequalities with parameters, we should first pay attention to whether it is necessary to have a classified discussion. If you encounter the following situations, you generally need to discuss: ① When multiplying and dividing a formula with parameters at both ends of an inequality, you need to discuss the positive, negative and zero properties of this formula; ② When monotonicity of exponential function and logarithmic function is needed in solving, their bases need to be discussed; (3) when the solution contains a letter.

Example: Solving the inequality X-A/X+ 1

Solution: arrange the deformation problem (a-1) x/a.

Discuss the coefficient of x by classification

( 1)when(a- 1)/a & gt； 0, that is, a

(2) When (a- 1)/a

(3) When (a- 1)/a=0, that is, a= 1, the inequality that X takes any real number holds.

function

1) Solution of analytic function: ① definition method (patchwork method): ② substitution method: ③ undetermined coefficient method: ④ assignment method: (2) Solution of function definition domain: The definition domain of parameter problem should be classified and discussed; For practical problems, after solving the resolution function; We must find its domain, and the domain at this time should be determined according to the actual meaning. (3) Solving the function value domain: ① Matching method: transforming it into a quadratic function and evaluating it by using the characteristics of the quadratic function; ② Inverse solution: X is represented by Y through the inverse solution, and then the value range of Y is obtained by solving the inequality; (4) Substitution method: transforming variables into functions of assignable fields and returning to ideas; ⑤ Triangular Bounded Method: Transform it into a function containing only sine and cosine, and use the boundedness of trigonometric function to find the domain; ⑥ Basic inequality method: use the average inequality formula to find the domain; ⑦ Monotonicity method: The function is monotonous, and the domain can be evaluated according to the monotonicity of the function. ⑧ Number-shape combination: According to the geometric figure of the function, the domain is found by the method of number-shape combination.

Properties of the function:

Monotonicity and Parity Monotonicity of Functions: Definition: Note that the definition is relative to a specific interval. The judgment methods are: difference comparison method and image method. Application: compare sizes, prove inequalities and solve inequalities. Parity: Definition: Pay attention to whether the interval is symmetrical about the origin, and compare the relationship between f(x) and f(-x). F (x)-f (-x) = 0f (x) = f (-x) f (x) is an even function; F (x)+f (-x) = 0f (x) =-f (-x) f (x) is odd function.

Example: f(x) is known as odd function, when x >; 0, f(x)=x( 1-x), then x

Solution: let x < 0, then-x >; 0 into f(x)=x( 1-x),

F (-x) =-x (1+x), and f (x) is odd function.

So f(-x)=-f(x) gives f(x)=x( 1+x),

I wrote this myself. If it is good, it can be adopted. (* _ _ *) ... Hee hee.