Summary of knowledge points of mathematical function in senior high school 1. For a set, we should grasp the "certainty, mutual difference and disorder" of the representative elements and elements of the set. What is the element? A represents the domain of the function y=lgx, b represents the range of values, and c represents the locus of points on the function 2. When performing intersection, union and complement operations on a set, don't forget the special case of the set itself and the empty set, and pay attention to the problem of the set with the help of the number axis and venn diagram. An empty set is a subset of all sets and the proper subset of all non-empty sets. Obviously, it is easy to solve A={- 1, 3} here. And b has only one element at most. So b can only be-1 or 3. According to the conditions, we can get a =- 1, a = 1/3. However, it should be noted here that there is another case where B is an empty set, that is, a=0. Don't forget. 3. Pay attention to the following properties: Know its source: If B is a subset of A, then the element a 1 has two choices (in or out). Similarly, for elements A2, A3, ... an, there are two choices, so there is always one choice, that is, set A has a subset. Of course, we should also pay attention to this situation, including the case where n elements do not exist, so the number of proper subset is, and the number of nonempty proper subset is (3) De Morgan's Law: some versions may be written like this, and you should be able to understand it after you meet them. 4. Will you solve the problem with the idea of complement set? (Exclusion method, indirect method). Note that sometimes you can get a lot of information from the set itself, so don't miss it when you do the problem; If I tell you that the function F (x) = AX2+BX+C (A > 0) When it monotonically decreases and monotonically increases, you should immediately know that the symmetry axis of the function is x= 1. In other words, when I say in the world, you should immediately think that m and n are actually two roots of the equation. 5. Know several forms of propositions, four forms of propositions and their relationships. A proposition with reciprocal negation is an equivalent proposition. The truth of the original proposition and the negative proposition; Whether it is an inverse proposition or not, a proposition is the same as true or false. 6, familiar with the nature of the necessary and sufficient conditions (college entrance examination) meet the conditions, meet the conditions, if; This is a sufficient and unnecessary condition; If; This is a necessary condition, but not a sufficient condition; If; Yes, a necessary and sufficient condition; If; This is neither sufficient nor necessary; 7. Do you know the concept of mapping? Mapping F: A → B, have you noticed the arbitrariness of elements in A and the uniqueness of corresponding elements in B? What kind of correspondence can constitute a mapping? (One-to-one, many-to-one, allowing the elements in B to have no original image. ) Pay attention to the solution of the number of mappings. If there are m elements in set A and n elements in set B, the number of mappings from A to B is nm. Such as: if,; Q: There are two mappings to and two mappings to; There is a function to, and if there is, there is a one-to-one mapping to. The number of intersection points between the function image and the straight line is. 8. What are the three elements of a function? How to compare whether two functions are the same? (Definition domain, corresponding rule, value domain) The judgment method of the same function: ① the expression is the same; 2 domain consistency (two points must be met at the same time) 9. What are the common types of function domains? Solve the domain of function: the denominator in the l fraction is not zero; L the number (or formula) under even roots is greater than or equal to zero; L Exponential radix is greater than zero and not equal to one; The base of L logarithmic formula is greater than zero, which is not equal to one, and the true number is greater than zero. L tangent function l cotangent function l inverse trigonometric function Y = Arcsinx has the domain of [- 1, 1], the domain of function y = arccosx is [- 1, 1], and the domain of function y = arcctgx is R. How to find the domain of compound function? The meaning domain is _ _ _ _ _ _ _ _. Solving the domain of compound function: if the domain is known, the domain can be solved by solving the value domain of x, and the domain of x is. For example, if the domain of a function is, the domain of is. Analysis: The domain of the function is known:; So there is something in it. Solution: According to the meaning of the question, the definition domain of ∴ is 1 1, and the solution of the function domain is 1. Direct observation method For some simple functions, the range can be obtained by observation. For example, find the range of the function y= 2. Matching method Matching method is one of the most basic methods to find the quadratic function value domain. For example, find the range of the function y= -2x+5, x [- 1, 2]. 3. Discriminant method can be used for quadratic function or fractional function (one of the numerator or denominator is quadratic), but this kind of problem can sometimes be simplified in other ways, so I don't have to stick to the upper and lower discriminant. I hope you can learn more about this type. 4. When it is difficult to find the range of the function directly by the inverse function method, the range of the original function can be determined by finding the definition range of the original function. For example, find the function y= range. 5. When it is difficult to find the range of the function by using the boundedness method of the function directly, the range of the function can be determined by using the boundedness of the learned function. What we call monotonicity is monotonicity of trigonometric functions. For example, find the range of the function y=,. 6. Monotonicity method of function is usually combined with derivative, which is one of the content examples of college entrance examination in recent years. Find the range of function y= (2≤x≤ 10). 7. method of substitution changed a function into a simple function through a simple method of substitution, and its question type is characterized by a formula model containing a root decomposition function or a trigonometric function. Method of substitution is one of the most important mathematical methods, and it also plays an important role in finding the range of functions. Find the range of function y=x+. The combination of 8 numbers and shapes has obvious geometric significance, such as the distance formula between two points and the slope of a straight line. If the combination of numbers and shapes is adopted, it will often be simpler and more pleasing to the eye. Example: Given that the point P(x.y) is on the circle x2+y2= 1, find the range of the function y=+. Solution: The original function can be simplified as: y=∣x-2∣+∣x+8∣ The above formula can be regarded as the sum of the distances from point P(x) to fixed points A(2) and B(-8) on the number axis. As can be seen from the ABove figure, when point P is on line segment AB, y=∣x-2∣+∣x+8∣=∣AB∣= 10. When point P is on the extension line or inverse extension line of line segment AB, y = ∣. +∞) Example Find the range solution of function y=+: the original function can be transformed into: y =+ The above formula can be regarded as the sum of the distances from point P(x, 0) on the X axis to two fixed points A(3, 2) and B(-2,-1). As can be seen from the figure, when point P is the intersection of a line segment and the X axis, y = Note: When calculating the sum of the two distances, we should use the basic inequalities a+b≥2, a+b+c≥3 (a, B, c∈) to find the maximum value of the function. When using the summation formula, the analytic formula of question characteristics requires the product to be constant, and the analytic formula requires the product to be constant, but sometimes we need to use techniques such as division, addition and square on both sides. Example:

Reciprocal method Sometimes, when you can't directly see the range of a function, you will find another situation. For example, there are many ways to find the range of a function y=. In a word, when you find the range of a function, you should first carefully observe the characteristics of its question type, and then choose the appropriate method. Generally speaking, direct method, function monotonicity method and basic inequality method are the main methods, and other special methods are considered.