The summary of formula is mainly the part of set function

The summary of formula is mainly the part of set function

Summary of knowledge points in each chapter of mathematics compulsory 1 in Senior High School
The first chapter is the concept of set and function
1、 The concept of set
1. The meaning of set: some specified objects together become a set, each of which is called an element
2. There are three characteristics of elements in a set
1. The certainty of elements; 2. The mutual dissimilarity of elements; 3. The disorder of elements
Description: (1) for a given set, the elements in the set are determined, and any object is or is not the element of the given set
(2) In any given set, any two elements are different objects. When the same object belongs to a set, it is only one element
(3) The elements in a set are equal and have no sequence, so to judge whether two sets are the same, we only need to compare whether their elements are the same, and do not need to check whether the sequence is the same
(4) The three characteristics of set elements make the set itself deterministic and holistic
3. Representation of set: { }Such as {basketball players of our school}, {Pacific Ocean, Atlantic Ocean, Indian Ocean, Arctic Ocean}
1. Use Latin letters to represent the set: a = {basketball players of our school}, B = {1,2,3,4,5}
2. Representation of set: enumeration and description
Note: common number sets and their notation:
The set of non negative integers (that is, the set of natural numbers) is denoted as: n
Positive integer set n * or N + integer set Z rational number set Q real number set R
On the concept of "belonging"
The elements of a set are usually represented by lowercase Latin letters. For example, if a is an element of a set, it is said that a belongs to a set, denoted as a ∈ A. on the contrary, a does not belong to a set, denoted as a? A
Enumeration: list the elements in the set one by one, and then enclose them with a brace
Description method: describes the common attributes of the elements in a collection, and writes them in braces to represent the collection. It uses certain conditions to indicate whether some objects belong to the collection
① Language Description: example: {triangle not right triangle}
② Mathematical expression description method: example: the solution set of inequality x-3 > 2 is {x? R | x-3 > 2} or {x | x-3 > 2}
4. Classification of sets:
1. A set of finite elements
2. A set of infinite elements
3. The set of empty sets without any elements, for example: {x | x2 = - 5}
2、 Basic relations between sets
1. Inclusion relation subset
Note: there are two possibilities: (1) a is a part of B; (2) a and B are the same set
Conversely: set a does not contain set B, or set B does not contain set a, denoted as a B or B a
2. "Equal" relation (5 = 5 if 5 ≥ 5 and 5 ≤ 5)
Example: let a = {x | x2-1 = 0} B = {- 1,1} "same elements"
Conclusion: for two sets a and B, if any element of set a is an element of set B, and any element of set B is an element of set a, we say that set a is equal to set B, that is: a = B
① Any set is a subset of itself
② Proper subset: if a í B and A1 B, then set a is the proper subset of set B, denoted as a B (or B a)
③ If a í B, B í C, then a í C
④ If a í B and B í a at the same time, then a = B
3. A set without any elements is called an empty set, denoted by Φ
Stipulation: an empty set is a subset of any set, and an empty set is a proper subset of any nonempty set
3、 Operation of set
1. The definition of intersection: generally, the set composed of all elements belonging to a and B is called the intersection of a and B
A ∩ B = {x | x ∈ a, and X ∈ B}
2. The definition of Union: generally, the union of a and B is composed of all elements belonging to a or B. It is denoted as a ∪ B (pronounced as "a and B"), that is, a ∪ B = {x | x ∈ a, or X ∈ B}
3. The properties of intersection and union: a ∩ a = a, a ∩ φ = φ, a ∩ B = B ∩ a, a ∪ a = a,
A∪φ= A ,A∪B = B∪A.
4. Complete set and complement
(1) Complement set: let s be a set, and a be a subset of S (that is, a set composed of all elements in s that do not belong to a), which is called the complement set (or coset) of subset a in S
It is recorded as: CSA, that is, CSA = {x | x? S and X? A}
S
CsA
A
(2) Complete set: if set s contains all the elements of each set we want to study, the set can be regarded as a complete set. It is usually represented by U
(3) Properties: 1) Cu (CUA) = a 2 (CUA) ∩ a = Φ 3 (CUA) ∪ a = u
2、 Some concepts of function
1. The concept of function: let a and B be nonempty sets of numbers. If, according to a certain corresponding relation F, there is a uniquely determined number f (x) corresponding to any number x in set a in set B, then f: a → B is a function from set a to set B. note: y = f (x), X ∈ A. where x is called an independent variable, The value range a of X is called the domain of function; the value of Y corresponding to the value of X is called the function value, and the set of function values {f (x) | x ∈ a} is called the domain of function
Note: 2 if only the analytic formula y = f (x) is given, and its definition field is not specified, then the definition field of function refers to the set of real numbers that can make the formula meaningful; 3 the definition field and value field of function should be written in the form of set or interval
Domain supplement
The set of real numbers x that can make the function meaningful is called the domain of function, The main basis for solving the system of inequalities in the domain of definition of a function is: (1) the denominator of the fraction is not equal to zero; (2) the number of even root is not less than zero; (3) the true number of the logarithm must be greater than zero; (4) the base of the exponent and logarithm must be greater than zero and not equal to 1. (5) if a function is composed of some basic functions through four operations, then, Its domain of definition is a set of values of X that make every part meaningful. (6) the base of exponent is not equal to zero. (6) the domain of definition of functions in practical problems should also ensure that the practical problems are meaningful
(also note: finding the solution set of inequality system is the domain of function.)
Three elements of a function: domain of definition, correspondence and range of values
Note again: (1) the three elements of a function are domain, correspondence and range of values. Since the range of values is determined by domain and correspondence, if the domain and correspondence of two functions are identical, they are called equal (or the same function) (2) two functions are equal if and only if their domain and correspondence are identical, It has nothing to do with the letters representing the values of independent variables and functions
(see related example 2 on page 21 of the textbook)
Range supplement
(1) The range of a function depends on its definition and corresponding rules. No matter what method is adopted, its definition should be considered first. (2) you should be familiar with the range of primary function, quadratic function, exponential function, logarithmic function and trigonometric function, which is the basis of solving the range of complex function
3. Function image knowledge induction
(1) Definition: in the plane rectangular coordinate system, the set C of points P (x, y) with X in function y = f (x), (x ∈ a) as abscissa and function value y as ordinate is called the image of function y = f (x), (x ∈ a)
The coordinates (x, y) of every point on C satisfy the functional relation y = f (x). Conversely, every group of ordered real numbers satisfying y = f (x) with X and y as coordinates (x, y) are all on C. that is, C = {P (x, y) | y = f (x), X ∈ a}
Image C is generally a smooth continuous curve (or straight line), or it may be composed of several curves or discrete points which have at most one intersection with any straight line parallel to y axis
(2) Painting
A. Point tracing method: according to the function analytic formula and definition field, find out some corresponding values of X and Y and list them. Take (x, y) as coordinates, trace the corresponding points P (x, y) in the coordinate system, and finally connect these points with smooth curves
B. Image transformation method (refer to compulsory 4 trigonometric function)
There are three common transformation methods: translation transformation, stretching transformation and symmetry transformation
(3) Function:
1. Intuitively see the nature of the function; 2, use the method of combination of number and shape to analyze the idea of solving problems and improve the speed of solving problems
Find out the mistakes in solving the problem
4. Understand the concept of interval quickly
(1) Classification of interval: open interval, closed interval, semi open and semi closed interval; (2) infinite interval; (3) number axis representation of interval
5. What is mapping
In general, let a and B be two nonempty sets. If according to a certain correspondence rule F, for any element X in the set a, there is a uniquely determined element Y corresponding to it in the set B, then the corresponding F: a B is said to be a mapping from the set a to the set B
Given a mapping from a set to B, if a ∈ a, B ∈ B. and element a corresponds to element B, then we call element B the image of element a, and element a the primitive image of element B
Note: function is a special mapping, and mapping is a special correspondence. ① set a, B and corresponding rule f are determined. ② corresponding rule has "directionality", that is, it emphasizes the correspondence from set a to set B, which is generally different from the correspondence from B to a. ③ for mapping f: a → B, it should meet the following requirements: (I) every element in set a has an image in set B, And the image is unique; (II) for different elements in set a, the corresponding image in set B can be the same; (III) every element in set B is not required to have an original image in set a
Common function representation and their respective advantages:
The graph of a function can be a continuous curve, a straight line, a broken line, a discrete point, etc, Pay attention to judge whether a graph is the basis of function image; 2. Analytic method: the definition field of function must be indicated; 3. Graphic method: point drawing method: determine the definition field of function; simplify the analytic formula of function; observe the characteristics of function; 4. List method: the selected independent variables should be representative and reflect the characteristics of definition field
Note: analytic method: easy to calculate the function value. List method: easy to find out the function value. Image method: easy to measure the function value
Supplement 1: piecewise function (see textbook p24-25)
Functions with different analytic expressions in different parts of the domain of definition. When calculating the value of a function in different ranges, the independent variable must be substituted into the corresponding expression. The analytic expression of a piecewise function can not be written into several different equations. Instead, several different expressions of the function value are written and enclosed in a left brace, (1) a piecewise function is a function, so don't mistake it for several functions; (2) the domain of a piecewise function is the union of the domains of each segment, and the range of values is the union of the domains of each segment
Supplement 2: composite function
If y = f (U), (U ∈ m), u = g (x), (x ∈ a), then y = f [g (x)] = f (x), (x ∈ a) is called the composite function of F and G
For example: y = 2sinx, y = 2cos (x2 + 1)
7. Monotonicity of functions
(1) . increasing function
Let the definition field of function y = f (x) be I, if for any two independent variables x1, X2 in an interval D in the definition field I, if x1

Please, 1, 5, 13, 25, 41 What is the general term formula of

5=1+4
13=1+4+8
25=1+4+8+12
41=1+4+8+12+16
An=1+4+8+12+16+..+4n=1+2n(n+1)

How to make the general term formula of 1,5,13,25 better?

In fact, the sequence given by LZ is a "second order arithmetic sequence", which is a kind of "higher order arithmetic sequence". The so-called second order arithmetic sequence takes the difference between the front and back terms of this sequence as the term of a new sequence. For example, take this problem as an example: {5-1,13-5,25-13} = {4,8,12} as arithmetic sequence, then we will take this sequence

If 3x + 2Y = 48.2x + y = 46. X = () y = ()

Simple substitution