Chapter collection
The concept of function, a collection of related concepts
The meaning of set: gather some specified objects into a set, where each object is called an object in the element set.
2. Three characteristics of elements in a set:
Uncertainty of elements; Anisotropy of two elements; 3。 Disordered element
Note: (1) For an element in a given set, it can be determined that no object or this is a given set of elements.
(2) At any given time, any two elements are different objects and can only be counted as one element in the same object set.
(3) The elements in the set are equal, and there is no order, so it is determined that the two sets are the same. It is only necessary to compare whether they have the same elements, and the order of inspection is not necessary.
(4) The set of elements with three characteristics is deterministic and comprehensive.
3. Assemble {...} {school basketball players} {Pacific Ocean, Atlantic Ocean, Indian Ocean, Arctic Ocean}
1。 The Latin letters of the collection are: A = {school basketball player}, B = {1, 2,3,4,5}.
2。 Representative set: legal enumeration.
Note: Symbols of commonly used number sets:
Non-negative integer (that is, natural number set), representing a positive integer: n
Set N * N+ of set, integer real number of set q? Diameter of rational number
The set elements of the concept of "belonging" are usually represented by lowercase Latin letters, such as the set of elements, which means that one belongs to a set and is represented by ∈ A, and vice versa, does one not belong to set A? Enumeration:
Enumerate the elements in the collection, and then use braces.
Description: Describes the common attributes, written braces and methods of the elements in the collection. Condition determines whether some objects belong to this collection.
(1) Language to describe the method: for example: {Triangle is not a right triangle}
(2) Mathematical formula method: Example: inequality X-3 >; 2 solution set {x? r | X-3 & gt； 2} or {x | x-3 >；; 2}
Collection classification:
1。 A finite set contains a finite set of elements.
2。 An infinite set contains an infinite number of elements of a set.
3。 When the empty set does not contain any element set: {X | x2 = -5}
collect
The basic relationship of 1 A subset of the "containing" relationship
Note: (1)A is a part of B; (2)A and B are the same set, and there are two possibilities.
Conversely, set A is not contained in set B, or set B does not contain set A, so it is marked as AB or BA.
2。 "Equality" (5≥5 and 5≤5, 5 = 5)
For example, let a = {x | x2-1= 0} b = {-1,1} the same element/>
Conclusion: For two sets A and B, if the element A of any group is the set element of B and the set B of any element is the element A of a set, we say that set A is equal to set B, that is, A = B.
(1) Any group is its own subset. Aia insurance
② proper subset: AIB, and A 1 B, then set A is the proper subset of set B, labeled AB (or BA).
③ If it is AIB·BIC, then it is AIC.
④ If AIB is BIA at the same time, then One = B.
A set that does not contain any elements is called an empty set and is denoted as φ.
It is stipulated that an empty set is a subset of any set and an empty set is an arbitrary non-empty proper subset.
Collect the calculation results. Definition of intersection: Generally speaking, all sets are composed of elements belonging to A and B, which are called intersection of A and B. ..
A∩B (pronounced "A horizontal B"), that is, A∩B = {X | X∈A and X∈B}.
2. Definition setting: Generally speaking, set A consists of all elements belonging to a set, or sets A and B belong to group B. It is expressed as: A∪B (pronounced as "A and B"), that is, A∪B = {X | X∈A, or x∈B}.
The nature of intersection and union of 3: A∩A = A, A∩φ=φ, A∩B = B∩A, A∪A = A,
/& gt； A∪φ= A，A∪B = B∪A。
Complement set of (1) complement set: Let s be a group and a subset of S (that is, elements that do not belong to a composite set are called subset A of S (I set it).
Vocabulary: CSA and CSA complement each other = {X | X? S and x? A}
Cyclosporin
one
(2) Complete Works: If all the elements in the set S are the sets we want to learn, then this set can be regarded as a complete project. Usually represented by u.
(3) Properties: (1) cu (cua) = a2 (cua) ∩ a = φ 3 (cua) ∪ a = u.
The concept of function
1。 Concept of function: Let A and B be non-empty number sets. If yes, there is a unique number in set B. Any set X of a certain number A corresponds to the function f(x) according to a certain correspondence F IT, and then the function F: A → B is set as A: Y = F(X) from set B. X. A set of values is called the function value X, and the value corresponding to the value Y? | X∈A} of {F(x) is called the functional domain of value.
Note: If only the analytical formula is given? = F(X), which has no specified domain. The function defined by this domain must make a meaningful formula set of real numbers, and the domain and value range of the three functions should be written in the form of a set or value range.
Add a custom field to a set of meaningful real numbers x, which is called a function in the function field. The main basis for finding the function definition domain is inequality: (1) the denominator of the fraction is not zero, (2) the quadratic root is not less than zero, (3) it must be the exponent of a real number greater than zero, (4) the base of the logarithm must be greater than zero and not equal to 1(5) the combination of four operations, then its definition domain is defined by a part of each value of X. (6) The definition domain of the function in practical problems must also be meaningful.
(Note: The set of inequality solutions found is the domain of function).
Three elements of a function: domain name, correspondence and scope.
Pay attention again to the corresponding relationship and scope of the domain of the ternary function composed of (1). Because the domain and correspondence of this range are certain, if the domain and correspondence of two functions are completely consistent, that is, the two functions are the same (or the same function), then (2) is equal to two functions, if and only if their domain and correspondence are completely the same, regardless of the values of the letters of the independent variables and functions. The function is the same: ① the judgment of expression is consistent; (2) Domain consistency (there are two requirements)
(see textbook 2 1)
Range of values? supplement
The function in the range of (1) depends on the domain and the corresponding laws to solve a series of values. The function of, should first consider the domain. (2) You should be familiar with the range of functions, such as linear function, quadratic function, exponential function, logarithmic function and trigonometric function, which is the range for solving complex functions.
Overview of function image knowledge
(1) Definition: In rectangular coordinate system, the set c of points P (x, y) with function y = f(x) (x∈A) as abscissa and function y as ordinate is called the image of function y = f(x) (x∈A).
The coordinates (x, y) of C at each point satisfy the functional relationship of y = f(x). Conversely, in order to satisfy the order of the real number Y = F (x) of each set at the point (x, y) of x and y coordinates, it is simply called C = {P(X, Y)| Y = F(X).
Figure. Generally, C is a smooth and continuous curve (or straight line), and it can also be composed of any curve or discrete points parallel to the intersection number of the most linear 1 on the Y axis.
(2) Painting
Plotting method: according to the resolution function, define the domain, find X, list some corresponding points P (X, y) delineated by the coordinates of the coordinate system with the corresponding Y value in (x, y), and finally connect these points with smooth curves.
B, image transformation method (see forced triangle has three functions)
Common transformation methods, namely translation transformation, expansion transformation and symmetry transformation.
(3) Function:
1, intuitively see the role of nature; Thoughts on solving problems by combining numbers with shapes. Improve the speed of solving problems.
Find mistakes and solve problems.
4。 Understand the concept of interval.
(1) classification: open interval, closed interval, semi-open and semi-closed interval (2) infinite interval; (3) The interval number axis represents.
5。 What is mapping?
Generally speaking, let A and B be two nonempty sets. If one determines the corresponding rule F element X, and any group only needs to determine the corresponding element Y in group B, then the corresponding telephone number: AB is the mapping from set B to set AB, and set AB is "F: A"
A set of mappings from A to B, ∈A, B∈B and element B corresponding to element A and element B are called elements, which are called the original images of similar elements of element B.
Description: This function is a special mapping, which is a special corresponding set A, B and the corresponding legal "orientation" to determine the corresponding rule F. It emphasizes the corresponding relationship between a set A and a set B, and generally it is different from B to A; ③ For the mapping f: a → b, it should be satisfied that the groups A and B in each element set in (i) are equal and unique; The different elements of a (ii) can be the same as the set corresponding to the set b (III) the original image of the set b of each element in the set is not required.
Common function laws and their respective advantages;
The function image of 1 can be a continuous curve, a straight line, a broken line or a discrete point. On the basis of careful observation, it is judged whether a graph is a function image. The analysis methods are as follows: the definition domain of the function must be specified; 3 mirror image method: we should pay attention to drawing with tracing point method: determine the functional characteristics of resolution function whose definition domain is simplified; 4 list method: the selected independent variables should be representative and reflect the characteristics of the field.
& lt/ Note: Analysis method is a function of calculated value. List method: it is easy to find the function value. Mirror method: function value that is easy to measure
Supplement: piecewise piecewise function (see textbook P24-25)
Different analytic expressions have different functions in different domain types. Function values must be independent variables and substituted into corresponding expressions in different ranges. In several different equations, analytical formulas can't be written. Write several expressions with different function values. And indicate the parameter values of each part respectively. The function is a function (1), so don't mistake it for multiple functions. (2) Sub-domain functional paragraph domain, whose range is set to the setting of paragraph value range.
Supplement: composite function
/& gt； If y = f (u) (∈ m) and U = G(X)(X∈A), then y = F [G(X)] = F(X), and (X∈A) is called the composite function of f and g.
For example: Y = 2sinX? = 2cos(X2 + 1)
7。 Monotonicity of function
& gt
(1) Improved functions
Let the domain I of the function y = f(x), and the domain D of any two of my independent variables x 1, x 2, when X 1
For any two independent variables, when x 1 F(X2), the value of d is x 1, x2, so the function F(X) in this time interval is a decreasing function. The interval d is y = f(x).
Note: The time interval of defined monotone function is called
Two independent variables X 1, X2, x 1
(2) the characteristics of the image
If the function is y = f(x), it is a increasing function or subtraction function in some time intervals, and the function y = (strictly monotone function f(x)). In this interval, the interval between monotone images increases from left to right, and the images decrease from left to right.
(3) The measurement method of monotonicity of monotone interval of function.
(1) Definition method:
Any X 1, X2, x 1
(b) Image method (looking at the elevator from the picture) _
(c) compound monotony
The monotonicity of the compound function f [G(X) is closely related to the monotonicity of its functions u = G(X) and Y = F(U), and its own laws are as follows:
function
Monotonous
U = G(X)
raise
raise
fewer/ lesser
negative
Is y = f(u)
BR/>； raise
raise
Very few
Y = F [G(X)] minus.
raise
negative
raise
Note: 1, monotone interval is only in the sub-interval of its definition domain, monotonicity is the same time interval and written together, and derivative method is set in elective learning to make simple decisions and memorize monotonously?
8。 functional check
(1) coupling function
Generally speaking, any one of x and (-x) within the definition of function f(x) is called the dual function of = f(x) in function f(x).
(2)。
Even-odd function, generally speaking, for an x of the function f(x) defined in any domain, if there is an =-function f(x) of (-x), then f(x) is called odd function.
Note: The function called by the function is odd function, or the function with even function parity is a function of the global nature of the function. There may be no parity, but it may be a function of even and odd numbers.
The necessary condition for the function shown in 2 to have parity is that x is defined in any range, and -X must be defined in a user-defined variable (that is, the symmetric origin in the domain of parity function). /a & gt；
(3) Image features have parity check function.
The bifunctional image is symmetrical about Y axis, and the odd function image is symmetrical about the origin.
The steps to define the parity format of the judgment function are: firstly, determine the definition domain of the function and whether the domain name is symmetrical; 2. Determine the relationship between f(X) and F(X); 3. Make a proper conclusion: if f(-x) =(x) or f(-x) -F(X)= 0, then F(X) is an even function; If f(-) =-F(X) or f(X)+ F(X)= 0, then F(X) is odd function.
Note: A necessary condition for the symmetric origin of function parity. First, whether the domain of a function is symmetric about the origin, and whether an asymmetric function is symmetric (1)(2)F(-x)= F(x) is sometimes difficult to determine, so we can consider whether f(-x) = 0 or function f (x)/(-).
9. Function of parsing expressions
The analytical expression of the function of (1) is a function, which requires the functional relationship between two variables. There are corresponding rules between them, which need the role of domain name.
(2) Expression and function analysis: undetermined coefficient method, an alternative method, omits bills. If the structure of function analysis is known, it is called the undetermined coefficient method of the expression of compound function F [G(x), which can be used for simple expressions with known value range when-$ $ is needed, and can also be used for scraping the common solution of known abstract function expressions to get F(X).
10。 Function (small) value (see equation elimination method on page 36 of the textbook for definition)
By using the image and the judgment function of the maximum monotonicity of the function with the (small) value of function 3, using the properties of quadratic function and the maximum (small) value of function (using method) 2: if the time interval [a, b] in function y = f(x) is monotonically increasing, in the time interval of monotonically decreasing function y [b, c] = function f(x),
Indexing function
Calculation of (1) exponent and power
1。 Radical concept: in general, if is called the second root (th root), where >; 1,∈*。
When it is an odd number, the nth root of a positive number is a positive number, and the nth root of a negative number is a negative number. At this time, the nth root symbol. This formula is called free radical (free radical), and the so-called root index (free radical index) here is called root (radical number).
When an even number is an n-th root, in this case, it is the inverse of two numbers. The nth square root of a positive number and the nth square root of a negative sign can be combined into one (>; 0)。 Don't even knot the roots several times; Any power root of 0 is 0, denoted as a >；;
Note: In odd numbers, in even numbers.
2。 Specific meanings of fractional exponential power and positive fractional exponential power
It is meaningless that the power of the positive fraction index is equal to 0 and the power of the negative fraction index is equal to 0.
The concept of meaning index of integer index is defined, and the operational properties of rational number index and integer index power can also be extended to rational number index power.
3。 Operational Properties of Exponential Power of Real Numbers
( 1); BR/>；
(2);
(3)。
Exponential function of (b) and its properties
1, the concept of exponential function: In general, this kind of function is called exponential function, where x is the independent variable and the function field is R.
Note: The cardinality of exponential function of the range cannot be negative, 0 and 1. Images and Properties of Exponential Functions
& gt
0 & lta & lt 1
/a & gt；
Properties of image feature function
/a & gt；
On the x and y axes are
Region diameter of negative infinite extension function
Image origin and y axis are asymmetrical.
Non-odd and non-even functions
Function diagrams, such as functions in the x-axis and range.
+
Fixed-point function image (0, 1)
Look from left to right.
An image that rises gradually from left to right.
The image is gradually declining.
Monotone decreasing function of increasing function;
The vertical axis of the image in the first quadrant is greater than 1.
The image of the first quadrant in the ordinate should not be much larger than 1.
Overall planning is less than 1.
In the second quadrant of the image in the second quadrant of the vertical axis, it is greater than 1.
BR/>；
The rising trend of the image is steeper and steeper.
The rising trend of images is getting slower and slower.
Function value? Start to slow down and grow rapidly;
The function value begins to drop rapidly, reaches a certain value, and then drops slowly;
Note: Using monotonicity and combining images, we can also see that:
(1) in [A, B], its range;
(2) Take all positive actions if and only if.
(3) Exponential function, total
(4) If,
, logarithmic function
( 1)
1。 Concept of number: Generally speaking, if the so-called concept of number, logarithm, is recorded as: (-base-real number-number)
Note: 1 Pay attention to the basic restrictions;
/& gt；
3 writing format of notes.
Two important logarithms:
Common Logarithm: Logarithm is10; BR/>；
Natural logarithm of 2: the base of irrational logarithm.
Logarithmic exponent between logarithm and exponent
←→ Based on electrical foundation >
←→ Real index → Power supply
(B) the operational nature of numbers
and
+;
2 - ;
3。
Note: the basic formula of change
(,and; )。
The basic formula of change draws the following conclusions (1), (2).
(2) Logarithmic function
1, the concept of logarithmic function: function and logarithmic function, which are independent variables, function domain (0, +∞).
Pay attention to the definition and form of similar exponential function and logarithmic function, and pay attention to the distinction.
: is not a logarithmic function, and can only be called as a logarithmic function.
2 Logarithmic functions are limited to:, and.
2, the essence of logarithmic function:
& gt
0 & lta & lt 1
Functional attribute
Image features
/a > of function image on y axis; Functional domain of (0, +∞)
The Origin of Image and the Asymmetry of Y Axis
Nonsingular non-even function
/& gt； The range of function values extends to the positive or negative infinite diameter in the Y-axis direction.
Fixed-point function image (1, 0)
/& gt； Image from left to right
Gradually increase from left to right, and see more and more functions.
The image is gradually declining and the function is reduced.
The overall picture of the first quadrant is greater than 0.
The overall picture of the first quadrant of is greater than 0.
The vertical axis of the second quadrant of the image is less than 0.
The overall picture of the second quadrant is less than 0.
(3) Power function
1, definition of power function: In general, the known form is a power function and a constant function.
2. Summarize the nature and function of power supply.
(1) definitions of all power functions (0, +∞) and image intersections (1,1);
> (2) The power function of the image passes through the origin and is an increasing function in time interval. Especially when the convex function of the image; At one time, the convex image of power function;
(3) The time interval of the power function image in the first quadrant decreases. When it tends to originate from the right side, the image in the axial direction is infinitely close to the image in the positive and semi-axis direction of the shaft, and infinitely close to the positive and semi-axis of the shaft side.
Functional application
Zeros of the roots of equations and functions
& gt 1, the concept of function zero: For functions, a real zero function should be established.
2. The meaning of zero function: the function of the image where the real root of the zero function equation intersects with the abscissa axis, namely:
The image of the intersection of the function of the equation and the real root axis and the function zero.
& lt/3, function zero method:
The demand function is zero:
(algebraic method) to find the real number of the root of the equation;
The properties of the equation of 2 (geometric method) can not find the zero point by linking with the function image and the function using the root formula.
4. Zero point of quadratic function:
Quadratic function.
1)△& gt； 0, the equation has two unequal real root images, two intersection points and the quadratic function of the axis, and the quadratic function has two zeros.
2) When △ = 0, the equation has two equal real roots (multiple roots), such as the intersection of the image and the axis of the quadratic function, and a quadratic function of double zero order or second order is zero.
3)△& lt； 0, the equation has no quadratic function with real roots, and there is no quadratic function with zero intersection with the axis.