[1. 1] Settings
The meaning and representation of the set 1. 1. 1
The concept of (1) set
The elements in a set are deterministic, different and out of order.
(2) Common number sets and their notation: n stands for natural number set, N* or n stands for positive integer set, z stands for integer set, q stands for rational number set, and r stands for real number set.
(3) the relationship between sets and elements
(4) Representation of sets
Natural language method: describe the scenery in the form of text narration.
Enumeration: Enumerate the elements in the collection one by one, and write them in braces to represent the collection.
(3) Description: the property of {x | x}, where x is the representative element of the set.
(4) Graphic method: use number axis or Wayne diagram to represent the set.
(5) Classification of sets
① A set with finite elements is called a finite set; ② A set with infinite elements is called an infinite set; ③ A set without any elements is called an empty set.
Basic relation between sets 1. 1.2
(6) Subset, proper subset and set are equal.
Basic operation of 1. 1.3 set
(8) Intersection, union and complement
The Solution of Inequality with Absolute Value and Quadratic Inequality in Supplementary Knowledge
The Absolute Solution of (1) Inequality
(2) Solving the quadratic inequality of one variable
[1.2] function and its representation
The concept of 1.2. 1 function
Conceptual function of (1) function
(1) Let A and B be two sets of nonempty numbers. If any number X in set A has a unique number f(x) corresponding to it in set B according to a certain corresponding rule F, then such correspondence (including the corresponding rules F of sets A and B and A to B) is called a function of sets A to B, and is denoted as F: A → B. 。
② Three elements of a function: domain, range and corresponding rules.
③ Only two domains and functions with the same corresponding rules are the same function.
(2) The concept and representation of interval
(3) When searching for the definition domain of a function, the following principles are generally followed:
① When f (x) is an algebraic expression, the domain is all real numbers.
② When f (x) is a fractional function, the domain is all real numbers whose denominator is not zero.
③ When f (x) is an even root and the modulus is non-negative, the domain is a set of real numbers.
④ The truth value of logarithmic function is greater than zero. When the base of logarithmic or exponential function contains variables, the base must be greater than zero and not equal to 1.
⑥ The base of zero (negative) exponential power cannot be zero.
⑦ If f(x) is a function composed of four operations of a finite number of basic elementary functions, its domain is generally the intersection of the domain of each basic elementary function.
⑧ For the problem of finding the domain of compound function, the general steps are as follows: If the domain of f(x) is known as [a, b], the domain of its compound function f[g(x)] should be solved by the inequality A ≤ g (x) ≤ B. 。
For the function with letter parameters, find its domain and discuss the letter parameters according to the specific situation of the problem.
⑩ The definition domain of the function determined by the actual problem should not only make the function meaningful, but also conform to the actual meaning of the problem.
(4) Find the range or maximum value of the function.
The common method of finding the maximum value of a function is basically the same as the method of finding the function value domain. In fact, if there is a minimum (maximum) number in the range of a function, this number is the minimum (maximum) value of the function. Therefore, the essence of finding the maximum value of a function is the same as that of the value domain, but the angle of asking questions is different. The common methods for finding the range and maximum value of a function are:
① Observation method: For relatively simple functions, we can directly get the range or maximum value through observation.
② Matching method: the resolution function is converted into the sum of the flat modulus with independent variables and the constant, and then the range or maximum value of the function is determined according to the range of variables.
④ Inequality method: Using basic inequality to determine the range or maximum value of the function.
⑤ Substitution method: Through variable substitution, complexity and difficulty can be simplified. Trigonometric substitution can transform the maximum problem of algebraic function into the maximum problem of trigonometric function.
⑥ Inverse function method: Determine the value range or maximum value of a function by using the reciprocal relationship between the definition range and the value range of the function and its inverse function.
⑦ Number-shape combination method: Determine the range or maximum value of a function by using function images or geometric methods.
The monotonicity of the function. Representation of function
(5) Representation method of functions
There are three commonly used methods to express functions: analytical method, list method and image method.
Analytic method is to express the corresponding relationship between two variables with mathematical expressions. List method is to express the corresponding relationship between two variables by list. Image method is to use images to express the corresponding relationship between two variables.
(6) The concept of mapping
Basic properties of [1.3] function
1.3. 1 monotonicity and maximum (minimum) value
Monotonicity of (1) Function
① Definition and determination method
(2) In the public domain, the sum of two increasing function is increasing function, the sum of two subtraction functions is subtraction function, increasing function minus one subtraction function is increasing function, and subtraction function minus one increasing function is subtraction function.
1.3.2 parity
(4) Functional equivalence
① Definition and determination method
(2) If the function f(x) is odd function and is defined as x=0, then f (0) = 0.
③ The symmetrical intervals of odd-numbered functions on both sides of the Y-axis are the same, while the symmetrical intervals of even-numbered functions on both sides of the Y-axis are opposite.
④ In the public domain, the sum (or difference) of two even functions (or odd function) is still an even function (or odd function), the product (or quotient) of two even functions (or odd function) is an even function, and the product (or quotient) of an even function and a odd function is odd function.
[Supplementary knowledge] Functional image
(1) drawings
Drawing by tracing points;
(1) Determine the functional domain; ② Analytical resolution function;
③ Discuss the properties of functions (parity and monotonicity); ④ Draw the image of the function.
Drawing with the transformation of basic function image;
It is necessary to accurately remember the images of various basic elementary functions such as linear function, quadratic function, inverse proportional function, exponential function, logarithmic function, power function and trigonometric function.
① Translation and transformation