The first chapter is the concept of set and function.

I. Collection of related concepts

1, meaning of set: some specified objects are set together into a set, and each object is called an element.

2. Three characteristics of elements in a set:

1. element determinism; 2. Mutual anisotropy of elements; 3. The disorder of elements

Description: (1) For a given set, the elements in the set are certain, and any object is either an element of the given set or not.

(2) In any given set, any two elements are different objects. When the same object is contained in a collection, it is only an element.

(3) The elements in the set are equal and have no order. So to judge whether two sets are the same, we only need to compare whether their elements are the same, and we don't need to examine whether the arrangement order is the same.

(4) The three characteristics of set elements make the set itself deterministic and holistic.

3. Representation of assembly: {…} such as {basketball players in our school}, {Pacific Ocean, Atlantic Ocean, Indian Ocean, Arctic Ocean}

1.Set is expressed in Latin letters: A={ basketball player of our school}, B={ 1, 2, 3, 4, 5}

2. Representation methods of sets: enumeration and description.

Note: Commonly used number sets and their symbols:

The set of nonnegative integers (i.e. natural number set) 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"

Elements in a collection are usually represented by lowercase Latin letters. For example, if A is an element of set A, it means that A belongs to set A, marked as A ∈ A; On the other hand, if a does not belong to the set a, it is marked as a? A

Enumeration: enumerate the elements in the collection one by one, and then enclose them in braces.

Description: A method of describing the common attributes of elements in a collection and writing them in braces to represent the collection. A method to indicate whether some objects belong to this set under certain conditions.

① Language Description: Example: {A triangle that is not a right triangle}

② Description of mathematical formula: Example: inequality X-3 >; The solution set of 2 is {x? r | x-3 & gt； 2} or {x | x-3 >；; 2}

4, the classification of the set:

1. The finite set contains a set of finite elements.

2. An infinite set contains an infinite set of elements.

3. An example of an empty set without any elements: {x | x2 =-5}

Second, the basic relationship between sets

1. "Inclusive" relation-subset

Note: There are two possibilities that A is a part of B (1); (2)A and B are the same set.

On the other hand, set A is not included in set B, or set B does not include set A, so it is recorded as A B or B A.

2. "Equality" relationship (5≥5, and 5≤5, then 5=5)

Example: let a = {x | x2-1= 0} b = {-1,1} "The elements are the same".

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. Aiya

② proper subset: If AíB and A 1 B, then set A is the proper subset of set B, and it is denoted as A B (or B A).

③ If aí b and bí c, then aí c.

④ If AíB and BíA exist at the same time, then a = b.

3. A set without any elements is called an empty set and recorded as φ.

It is stipulated that an empty set is a subset of any set and an empty set is a proper subset of any non-empty set.

Third, the operation of the set.

Definition of 1. intersection: Generally speaking, the set consisting of all elements belonging to A and B is called the intersection of A and B. 。

Write A∩B (pronounced "A to B"), that is, A∩B={x|x∈A, x ∈ b}.

2. Definition of union: Generally speaking, a set consisting of all elements belonging to set A or set B is called union of A and B. Note: A∪B (pronounced as "A and B"), that is, A∪B={x|x∈A, or x ∈ b}.

3. The nature 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 works and supplements

(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 complement set) of subset A in S..

Note: CSA is CSA ={x | x? S and x? A}

S

CsA

A

(2) Complete Works: If the set S contains all the elements of each set we want to study, this set can be regarded as a complete set. Usually represented by u.

(3) Properties: (1) cu (cua) = a2 (cua) ∩ a = φ 3 (cua) ∪ a = u.

Second, the related concepts of function

The concept of 1. function: Let a and b be non-empty number sets. If any number X in set A has a unique number f(x) corresponding to it according to a certain correspondence F, then F: A → B is called a function from set A to set B, and is denoted as y=f(x). 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 range of the function.

Note: 2 If only the analytical formula y=f(x) is given without specifying its domain, the domain of the function refers to the set of real numbers that can make this formula meaningful; The definition and range of functions should be written in the form of sets or intervals.

Domain supplement

The set of real numbers x that can make a function meaningful is called the domain of the function. The main basis for finding the domain of function is that the denominator of (1) score is not equal to zero; (2) The number of even roots is not less than zero; (3) The truth value of the logarithmic formula must be greater than zero; (4) Exponential radix and logarithmic radix 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 is a set of values of x that make all parts meaningful. (6) Exponential radix cannot be equal to zero. (6) The definition domain of function in practical problems should also ensure that practical problems are meaningful.

(Also note that finding the solution set of inequality group is the domain of function. )

The three elements of a function: definition domain, correspondence relationship and value domain.

Note again: (1) The three elements that make up a function are domain, correspondence and value. Because the range is determined by the domain and the corresponding relationship, two functions are called equal (or the same function) if and only if their domain and the corresponding relationship are exactly the same, but the independent variables and function values are represented by letters. The judgment method of the same function: ① the expressions are the same; (2) Domain consistency (two points must be met at the same time)

(See related example 2 on page 2 1 of the textbook)

Value range supplement

(1), the range of a function depends on the defined range and the corresponding law. No matter what method is adopted to find the range of a function, its defined range should be considered first. (2) The range of linear function, quadratic function, exponential function, logarithmic function and trigonometric function should be familiar, which is the basis for solving the numerical range of reply.

3. Function image knowledge induction

(1) Definition: In the plane rectangular coordinate system, the set c of points P(x, y) with functions y = f (x) and (x ∈ a) as abscissa and function value y as ordinate is called the image of functions y = f (x) and (x ∈ a).

The coordinates (x, y) of each point on c satisfy the functional relationship y=f(x). On the other hand, the points (x, y) whose coordinates are x and y for each group of ordered real numbers satisfying y=f(x) are all on c, that is, c = {p (x, y) | y = f (x).

Image C is generally a smooth and continuous curve (or straight line), or it may be composed of several curves or discrete points, and it has at most one intersection with any straight line parallel to the Y axis.

(2) Painting

A. Point tracing method: according to the resolution function and the definition domain, find some corresponding values of x and y and list them, trace the corresponding points p (x, y) in the coordinate system with (x, y) as coordinates, and finally connect these points with smooth curves.

B, image transformation method (please refer to the compulsory 4 trigonometric function)

There are three commonly used transformation methods, namely translation transformation, expansion transformation and symmetry transformation.

(3) Function:

1, intuitively see the nature of the function; 2. Analyze the thinking of solving problems by combining numbers and shapes. Improve the speed of solving problems.

Find mistakes in solving problems.

4. Understand the concept of interval.

Classification of (1) interval: open interval, closed interval and semi-open and semi-closed interval; (2) Infinite interval; (3) The number axis representation of the interval.

5. What is mapping?

Generally speaking, let A and B be two nonempty sets. If any element X in set A has a unique element Y corresponding to it according to a corresponding rule F, then the corresponding F: A B is the mapping from set A to set B ... Write it as "f: a b"

Given a mapping from set a to set b, if A ∈ A, B ∈ B and element a correspond to element b, then we call element b the image of element a and element a the original 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 definite; (2) The correspondence rule is directional, 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 be satisfied that: (i) every element in set A has an image in set B, and the image is unique; (ii) Different elements in set A and corresponding images in set B can be the same; (iii) Each element in set B does not need to have an original image in set A. ..

Common function representations and their respective advantages;

The 1 function image can be a continuous curve, a straight line, a broken line, a discrete point, etc. Pay attention to the basis of judging whether a graph is a function image; 2 analytical method: the domain of the function must be specified; 3 mirror image method: attention should be paid to drawing by tracing point method: determine the definition domain of function; Simplify the analytical formula of the function; Observe the characteristics of the function; List method: the selected independent variables should be representative and reflect the characteristics of the field.

Note: Analytical method: it is convenient to calculate the function value. List method: it is easy to find the function value. Mirror image method: convenient to measure function value

Supplement 1: piecewise function (see textbook P24-25)

There are different functions in different parts of the domain to parse the expression. When finding the function value in different ranges, the independent variable must be substituted into the corresponding expression. The analytic expression of piecewise function cannot be written as several different equations. Instead, write several different expressions of function values and enclose them in left brackets, indicating the values of independent variables of each part respectively. (1) piecewise function is one function, so don't mistake it for several functions. (2) The definition domain of piecewise function is the union of the definition domain of each segment, and the value domain is the union of the value domain 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). Incremental function

Let the domain of function y=f(x) be I, if for any two independent variables x 1, x2 is in an interval d within the domain I, when X 1

If the values of any two independent variables in the interval d are both x 1, x2, when X 1 f (x2), then f (x) is said to be a decreasing function in this interval. The interval d is called monotonically decreasing interval y=f(x).

Note: the monotonicity of 1 function is a property in a certain interval within the definition domain and a local property of the function;

2 must be any two independent variables x 1, and x2 is in the interval d; When x 1

(2) the characteristics of image

If the function y=f(x) is increasing function or subtraction function in a certain interval, it is said that the function y=f(x) has (strict) monotonicity in this interval, and the image of increasing function rises from left to right, and the image of subtraction function falls from left to right.

(3) The method of judging monotone interval and monotonicity of function.

(1) Definition method:

1 is x 1, x2∈D, X 1

(b) Image method (rising and falling from image) _

(c) Monotonicity of composite functions

The monotonicity of the compound function f[g(x)] is closely related to the monotonicity of its constituent functions u=g(x) and y=f(u), and its laws are as follows:

function

monotonicity

u=g(x)

raise

raise

negative

negative

y=f(u)

raise

negative

raise

negative

y=f[g(x)]

raise

negative

negative

raise

Note: 1, the monotone interval of the function can only be a sub-interval of its domain, and the intervals with the same monotonicity cannot be summed together to write its union. 2. Do you remember the simple derivative method of judging monotonicity that we learned in elective courses?

8. Parity of functions

(1) even function

Generally speaking, f (-x) = f(x) exists for any x in the domain of function f(x), so f (x) is called even function.

(2) odd function

Generally speaking, F (-x) =-f(x) exists for any X in the definition domain of function f(x), so F (x) is called odd function.

Note: 1 function is odd function or even function is called function parity, and the parity of function is the global property of function; A function may have no parity, or it may be both a odd function and an even function.

According to the definition of function parity, a necessary condition of function parity is that -x must also be an independent variable in the definition domain for any x in the definition domain (that is, the definition domain is symmetrical about the origin).

(3) Features of images with parity function

The image of even function is symmetrical about y axis; Odd function's image is symmetrical about the origin.

Summary: Use the defined format to judge the parity of the function. Steps: 1 First, determine the domain of the function and judge whether its domain is symmetrical about the origin; 2 determine the relationship between f (-x) and f(x); 3. It is concluded that if f (-x) = f(x) or f (-x)-f (x) = 0, then f(x) is an even function; If f (-x) =-f(x) or f (-x)+f (x) = 0, then f(x) is odd function.

Note: the symmetry of the function definition domain about the origin is a necessary condition for the function to have parity. First, whether the domain of the function is symmetric about the origin, and if not, whether the function is odd or even. If it is symmetric, (1) will be judged according to the definition. (2)f(-x)= f(x) is sometimes difficult to judge. We can consider whether f (-x) f (x) = 0 or F (x)/F (-x) =1; (3) Using the image judgment of theorems or functions.

9. Analytic expression of function

(1). The analytic formula of the function is a representation of the function. When the functional relationship between two variables is needed, the corresponding law between them and the definition domain of the function are needed.

(2) The main methods to find resolution function are: undetermined coefficient method, method of substitution method, parameter elimination method, etc. If the structure of the resolution function is known, the undetermined coefficient method can be used; When the expression of the compound function f[g(x)] is known, method of substitution can be used, so we should pay attention to the value range of elements; When the known expression is simple, the matching method can also be used; If the expression of abstract function is known, f(x) is usually obtained by solving equations and eliminating parameters.

10. Maximum (minimum) value of the function (see textbook p36 for definition).

1 Find the maximum (minimum) value of the function by using the properties of quadratic function (matching method) 2 Find the maximum (minimum) value of the function by using the image 3 Judge the maximum (minimum) value of the function by using the monotonicity of the function: If the function y=f(x) monotonically increases in the interval [a, b] and monotonically decreases in the interval [b, c],

Chapter II Basic Elementary Functions

I exponential function

(A) the operation of exponent and exponent power

The concept of 1. radical: generally, if, then it is called n-degree radical, where >: 1 and ∈ *.

When it is an odd number, the power root of a positive number is a positive number and the power root of a negative number is a negative number. At this point, the power root of is represented by a symbol. The formula is called radical, here it is called radical index, here it is called radical.

When it is an even number, a positive number has two power roots, and the two numbers are opposite. At this time, the positive power roots of positive numbers are represented by symbols, and the negative power roots are represented by symbols. Positive and negative power roots can be combined into +(>: 0). It can be concluded that negative numbers have no even roots; Any power root of 0 is 0, which is recorded as.

Note: In odd numbers, even numbers,

2. Power of fractional exponent

The meaning of the power of the positive fractional index stipulates:

,

A positive fractional exponent power of 0 is equal to 0, and a negative fractional exponent power of 0 is meaningless.

It is pointed out that after defining the meaning of fractional exponent power, the concept of exponent is extended from integer exponent to rational exponent, and the operational nature of integer exponent power can also be extended to rational exponent power.

3. Operational Properties of Exponential Power of Real Numbers

( 1) ;

(2) ;

(3) .

(B) Exponential function and its properties

1, the concept of exponential function: Generally speaking, a function is called an exponential function, where x is the independent variable and the domain of the function is R. 。

Note: The base range of exponential function cannot be negative, zero 1.

2. Images and properties of exponential function

a & gt 1

0 & lta & lt 1

Image characteristics

Functional attribute

Infinitely extending in the positive and negative directions of the x axis and the y axis.

The domain of a function is r.

The image is asymmetric about the origin and y axis.

Nonsingular non-even function

Functional images are all above the x axis.

The range of the function is R+

Function images all pass through the fixed point (0, 1)

Looking from left to right,

The image is gradually rising.

Looking from left to right,

The image is gradually declining.

increasing function

Descending function

The vertical coordinates of the images in the first quadrant are all greater than 1.

The vertical coordinates of the images in the first quadrant are all less than 1.

The vertical coordinates of the images in the second quadrant are all less than 1.

The vertical coordinates of the images in the second quadrant are all greater than 1.

The rising trend of the image is steeper and steeper.

The upward trend of image is getting slower and slower.

The function value begins to grow slowly, and then grows rapidly after reaching a certain value;

The function value begins to decrease rapidly, and then decreases slowly after reaching a certain value.

Note: Using the monotonicity of the function and combining with the image, we can also see that:

(1) on [a, b], the range is or;

(2) If yes, then; Take all positive numbers if and only if;

(3) For exponential function, there is always;

(4) When, if, then;

Second, the logarithmic function

(1) logarithm

The concept of 1. Logarithm: Generally speaking, if, then this number is called logarithm with base, written as: (-base,-true number,-logarithmic formula).

Note: 1 Pay attention to the limit of cardinality, and;

2 ;

Pay attention to the writing format of logarithm.

Two important logarithms:

Common logarithm of 1: logarithm based on 10;

Natural logarithm: Logarithm based on irrational numbers.

Conversion between Logarithmic and Exponential Expressions

Logarithmic exponential expression

Logarithmic radix → power radix

Logarithm-→ exponent

Real number ←→ power

(B) the operational nature of logarithm

If,, and,, then:

1 + ;

2 - ;

3 .

Note: Bottom-changing formula

(,and; And; ).

The following conclusions (1) are derived by using the formula of changing the bottom; (2) .

(2) Logarithmic function

1, the concept of logarithmic function: function, also called logarithmic function, where is the independent variable and the domain of the function is (0, +∞).

Note: The definition of logarithmic function of 1 is similar to that of exponential function, and both are formal definitions. Pay attention to discrimination.

For example, none of them are logarithmic functions, only logarithmic functions.

2 limits of logarithmic function to base:, and.

2, the nature of the logarithmic function:

a & gt 1

0 & lta & lt 1

Image characteristics

Functional attribute

Functional diagrams are all on the right side of the y axis.

The domain of the function is (0, +∞)

The image is asymmetric about the origin and y axis.

Nonsingular non-even function

Extend infinitely in the positive and negative directions of the y axis.

The range of the function is r.

Function images all pass through fixed points (1, 0).

Looking from left to right,

The image is gradually rising.

Looking from left to right,

The image is gradually declining.

increasing function

Descending function

The image ordinate of the first quadrant is greater than 0.

The image ordinate of the first quadrant is greater than 0.

The vertical coordinates of the images in the second quadrant are all less than 0.

The vertical coordinates of the images in the second quadrant are all less than 0.

(3) Power function

1. Definition of power function: Generally speaking, a shape function is called a power function, where is a constant.

2. Summarize the properties of power function.

(1) All power functions are defined at (0, +∞), and the image passes through (1,1);

(2) When the image of the power function crosses the origin, it is an increasing function in the interval. Especially, when the image of power function is convex; When the image of the power function is convex;

(3) The image of power function is a decreasing function in the interval. In the first quadrant, when moving from the right to the origin, the image is infinitely close to the positive semi-axis of the shaft on the right side of the shaft, and infinitely close to the positive semi-axis of the shaft above the shaft when moving to the origin.

Chapter III Functional Application

First, the root of the equation and the zero of the function.

1, the concept of function zero: for a function, the real number that makes it true is called the zero of the function.

2. The meaning of the zero point of the function: the zero point of the function is the real root of the equation, that is, the abscissa of the intersection of the image of the function and the axis. Namely:

The equation has a real root function, the image has an intersection with the axis, and the function has a zero point.

3, the role of zero solution:

Find the zero point of a function:

1 (algebraic method) to find the real root of the equation;

2 (Geometric method) For the equation that can't be solved by the root formula, we can relate it with the image of the function and find the zero point by using the properties of the function.

4. Zero point of quadratic function:

Quadratic function.

1) △ > 0, the equation has two unequal real roots, the image of the quadratic function has two intersections with the axis, and the quadratic function has two zeros.

2) △ = 0, the equation has two equal real roots (multiple roots), the image of the quadratic function intersects with the axis, and the quadratic function has a double zero or a second-order zero.

3) △ < 0, the equation has no real root, the image and axis of the quadratic function have no intersection, and the quadratic function has no zero point.