The first chapter is the concept of set and function.
I. Collection of related concepts
The meaning of 1. set
2. Three characteristics of elements in a set:
The certainty of (1) element is as follows: the highest mountain in the world.
(2) The mutual anisotropy of elements, such as the set of happy letters {H, a, p, Y}.
(3) The disorder of elements: for example, {a, b, c} and {a, c, b} represent the same set.
3. Representation of assembly: {…} For example, {basketball players in our school}, {Pacific Ocean, Atlantic Ocean, Indian Ocean, Arctic Ocean}
(1) The set is expressed in Latin letters: A={ basketball players in our school}, B={ 1, 2, 3, 4, 5}.
(2) Representation of sets: enumeration and description.
Note: Commonly used digit sets and their symbols:
The set of nonnegative integers (i.e. natural number set) is recorded as n.
Positive integer set N* or N+ integer set z rational number set q real number set r
1) enumeration: {A, b, C...}
2) Description: A method of describing the common attributes of elements in a collection and writing them in braces to represent the collection. {x？ r | x-3 & gt； 2}，{ x | x-3 & gt； 2}
3) Language description: Example: {A triangle that is not a right triangle}
4) Venn diagram:
4, the classification of the set:
The (1) 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: A=B (5≥5 and 5≤5, then 5=5)
Example: let a = {x | x2-1= 0} b = {-1,1} "Two sets are equal if their elements are the same".
Namely: ① Any set is a subset of itself. Answer? A
② proper subset: If a? B and a? B then says that set A is the proper subset of set B, and writes it as A B (or B A).
3 if a? B，B？ C, then a? C
4 if a? 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.
A set of n elements, including 2n subsets and 2n- 1 proper subset.
Third, the operation of the set.
Complement set of intersection and union of operation types
Defined by all elements belonging to A and B, it is called the intersection of A and B, and marked as A B (pronounced' A crosses B'), that is, AB = {X | X A, and X B}.
A set consisting of all elements belonging to set A or set B is called the union of A and B. Note: A B (pronounced as' A and B'), that is, A B ={x|x A, or X B}).
Let S be a set, A is a subset of S, and the set of all elements in S that do not belong to A is called the complement (or complement) of subset A in S.
Remember, that's
CSA=
leather
elegant
draw
show
nature
Quality A A=A
aφ=φ
A B=B A
one
A B B
A A=A
Aφ= A
A B=B A
one
A B B
(CuA) (cub)
= copper (boron)
(CuA) (cub)
= copper (boron)
A (CuA)=U
a(CuA)=φ。
Example:
1. The following four groups of objects can form a set ().
A all the tall students in a class, B famous artists, C all the big books, the reciprocal of D is equal to its own real number.
2. Set {a, B, c} has a proper subset.
3. If the set m = {y | y = x2-2x+ 1, x r} and n = {x | x ≥ 0}, the relationship between m and n is.
4. let A= and B=. If it is B, the value range of is
5.50 students did two kinds of experiments: physics and chemistry. It is known that 40 students have done physics experiments correctly and 365,438+0 students have done chemistry experiments correctly.
If four people do both experiments wrong, then two people do both experiments right.
6. Represents the set M=. Descriptive method consists of points in the shadow part of the figure (including points on the boundary).
7. Given the set A∩C =φ{ x | x2+2x-8 = 0}, b = {x | x2-5x+6 = 0}, c = {x | x2-MX+m2- 19 = 0}, if b ∩.
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:
1. domain: the set of real numbers x that can make the function meaningful is called the domain of the function.
The main basis for finding the domain of function is:
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) The bases of exponential and logarithmic expressions must be greater than zero and not equal to 1.
(5) If a function is a combination of some basic functions through four operations, then its domain is a set of values of x that make all parts meaningful.
(6) The index is zero, and the bottom cannot be equal to zero.
(7) The definition domain of the function in the actual problem should also ensure that the actual problem is meaningful.
The judgment method of the same function: ① the expression is the same (regardless of letters representing independent variables and function values); (2) Domain consistency (two points must be met at the same time)
(See related example 2 on page 2 1 of the textbook)
2. Range: consider its definition range first.
(1) observation method
(2) Matching method
(3) substitution method
3. Function image knowledge induction
(1) Definition: The set c of points P(x, y) with functions y=f(x) and (x ∈ a) as abscissa and function value y as ordinate in a plane rectangular coordinate system is called the image of functions y=f(x) and (x ∈ a).
(2) Painting
First, the tracking method:
B, image transformation method
There are three common conversion methods.
1) translation transformation
2) Telescopic transformation
3) Symmetric transformation
4. 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.
map
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 ... written down as "F (corresponding): A (original image) B (image)"
For mapping f: a → b, then:
(1) Every element in set A has an image in set B, and the image is unique;
(2) Different elements in set A, and the corresponding images in set B can be the same;
(3) Each element in set B does not need to have an original image in set A. ..
6. Piecewise function
(1) Functions with different analytic expressions in different parts of the domain.
(2) The values of the independent variables of each part.
(3) The domain of piecewise function is the intersection of the domain of each segment, and the range is the union of the range of each segment.
Supplement: composite function
If y = f (u) (u ∈ m) and u = g (x) (x ∈ a), then y=f[g(x)]=F(x)(x∈A) is called the composite function of f and g.
Two. Properties of functions
Monotonicity of 1. Function (Local Property)
(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 a function is a local property of the function;
(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 Let x 1, x2∈D, x 1
2 difference f (x1)-f (x2);
3 deformation (generally factorization and formula);
○○○○○○○○○○○○○○○○○○○○○○○○○○○○○967
○5 Draw a conclusion (point out the monotonicity of the function f(x) in the given interval d).
(b) Image method (looking up and down from the 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 law is "same increase but different decrease".
Note: The monotone interval of a function can only be a subinterval of its domain, and the intervals with the same monotonicity cannot be summed together to write its union.
8. Parity of function (global property)
(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.
(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.
Judging the parity of a function according to its definition;
○ 1 First, determine the definition domain of the function and judge whether it is symmetrical about the origin;
2 determine the relationship between f (-x) and f(x);
○3 The corresponding conclusion is drawn: 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: Symmetry of the definition domain of a function 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) judging by 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 for finding analytic expressions of functions are:
1) matching method
2) undetermined coefficient method
3) Alternative methods
4) Parameter elimination method
10. Maximum (minimum) value of the function (see textbook p36 for definition).
○ 1 Use the property of quadratic function (collocation method) to find the maximum (minimum) value of the function.
○2 Use images to find the maximum (minimum) value of the function.
○3 Use monotonicity of function to judge the maximum (minimum) value of function;
If the function y=f(x) monotonically increases in the interval [a, b] and monotonically decreases in the interval [b, c], then the function y=f(x) has the maximum value f (b) at x=b;
If the function y=f(x) monotonically decreases in the interval [a, b] and monotonically increases in the interval [b, c], then the function y=f(x) has a minimum value f (b) at x=b;
Example:
1. Find the domain of the following function:
⑴ ⑵
2. Let the domain of the function be _ _
3. If the domain of the function is, the domain of the function is.
4. Function, if, then =
5. Find the scope of the following functions:
⑴ ⑵
(3) (4)
6. Know the function and find the analytical formula of the function.
7. If the known function is satisfied, then =.
8. Let it be odd function on R, if, then if =
The analytical formula of r is
9. Find the monotone interval of the following functions:
⑴ ⑵ ⑶
10. Judge the monotonicity of the function and prove your conclusion.
1 1. Set a function to judge its parity and prove:
Chapter II Basic Elementary Functions
I exponential function
(A) the operation of exponent and exponent power
The concept of 1. radical: generally, if, it is called the second radical of, where >; 1 and ∈ *.
Negative numbers have no even roots; Any power root of 0 is 0, which is recorded as.
When it is odd, when it is even,
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.
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 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 & lt； a & lt 1
Domain r domain r
Range y > 0 range y > 0
It increases monotonically on R and decreases monotonically on R.
Nonsingular non-even function
All function images pass through a fixed point (0, 1). All function images pass through a fixed point (0, 1).
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;
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 cardinal number, and;
○2 ;
3 pay attention to the writing format of logarithm.
Two important logarithms:
○ 1 common logarithm: logarithm based on 10;
○2 Natural Logarithm: Logarithm based on irrational numbers.
Reciprocity of exponential formula and logarithmic formula
Power real number
= N = b
cardinal number
Exponential logarithm
(B) the operational nature of logarithm
If,, and,, then:
○ 1 ? + ;
○2 - ;
○3 .
Note: Bottom-changing formula
(,and; And; ).
The following conclusions are drawn by using the formula of changing the bottom.
( 1) ; (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 ○ 1 logarithmic function is similar to that of exponential function, both of which are formal definitions. Pay attention to discrimination. For example, none of them are logarithmic functions, only logarithmic functions.
○2 restrictions of logarithmic function on cardinality:, and.
2, the nature of the logarithmic function:
a & gt 1 0 & lt； a & lt 1
Domain x > 0 domain x > 0
The range of values is r, and the range of values is R.
Increase on r, decrease on r.
Function images all pass through the fixed point (1, 0). Function images all pass through the fixed point (1, 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.
Example:
1. known A >；; 0, a 0, the image with function y=ax and y=loga(-x) can only be ().
2. Calculation: ①; ② = ; = ;
③ =
3. The decreasing interval of the function y=log (2x2-3x+ 1) is
4. If the maximum value of the function in the interval is 3 times the minimum value, then a=
5. The domain of (1) and the range of (2) are known.
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.
That is, the image of the real root function of the equation has an intersection with the axis, and the function has a zero point.
3, the role of zero solution:
○ 1 (algebraic method) to find the real root of the equation;
○2 (Geometric method) For the equation that cannot be solved by the root formula, it can be linked with the image of the function, and the zero point can be found 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, the image of the quadratic function intersects with the axis, and the quadratic function has double zeros or second-order zeros.
(3) △ < 0, the equation has no real root, the image of the quadratic function has no intersection with the axis, and the quadratic function has no zero point.
5. Functional model
Can you solve your problem?