Knowledge structure:

function

I. Gathering knowledge:

Basic concepts of 1. set

The whole of some specific objects is called a collection. Every object in a collection is called an element of this collection. A is an element of set A, which is denoted as a∈A, while A is not an element of set A, which is denoted as A? A.

The nature of (1) set: for a given set, its elements are deterministic, different and out of order.

(2) Representation of set: The representation methods of set include enumeration method, description method and graphic method.

(3) Common number sets are: n (natural number set), N* or N+ (positive integer set), z (integer set) and q (rational number.

Set), r (real number set), c (complex number set)

2, the nature of the set ① Any set is a subset of itself, recorded as; (2) An empty set is a subset of any set, denoted as;

③ An empty set is the proper subset of any non-empty set; If, at the same time, then a = b.

If.

[Note]: ①Z = {integer }(√) Z ={ all integers} (×)

3.① {(x, y)|xy =0, x∈R, y∈R} point set on the coordinate axis. ②{(x, y) | xy < 0, x∈R, y∈R points are in two or four quadrants.

(3) {(x, y) | xy > 0, x∈R, y∈R} point sets of one and three quadrants.

Example: solution set {(2, 1)}.

② The intersection of point set and number set is. (Example: A ={(x, y) | y = x+1} b = {y | y = x2+1} then A∩B =)

4.①n elements have 2n subsets. ②n elements have 2n- 1 proper subset. ③ There are 2n-2 nonempty proper subset with n elements.

5. The relationship between set and set

( 1) ①A？ B is defined as: any a∈A has A ∈ B.2A = B? Answer? B and b? A.

(2) a ∩ b = {x ∣ x ∈ a and x ∈ b}.

(3) a ∪ b = {x ∣ x ∈ a or x ∈ b}.

(4)? A={x∣x University? A and x∈U (where u is a complete set, the same below).

6. The nature of intersection, union and complement of sets

( 1)A∩？ =? a∨？ =A A∩(？ U A)=？ A∩U=A A∪U=U AU(？ U A)=U？ u(？ U a) = a (where? Is an empty set).

(2)A∩B = B∩A A A∪B = B∪A。

(3)(A∩B)∩C=A∩(B∩C)，(A∪B)∪C=A∪(B∪C)。

(4) if a? B, then a ∩ b = a, a ∪ b = B

(5)A∩(B∪C)=(A∪B)∪( A∪C)A ∪( B∪C)=(A∪B)∪( A∪C)。

(6)? U(A ∩B)=(？ u A)∨(？ U B)，？ U(A∪B)=(？ U A)∩(？ U B)。

Second, the concept of function:

1. Functional concept

Given two groups of nonempty numbers A and B, if any number X in A corresponds to a unique number Y according to a certain correspondence F, then the correspondence F is called a function defined on A, and denoted as F; A→B or y = f (x), x ∈ a. At this time, x is called the independent variable, set A is called the domain of the function, and set C = {f (x) | x ∈ a} is called the range of the function, c? A function has three elements: domain, range and correspondence.

2. Representation of functions

List method: a method of expressing the functional relationship between two variables in tabular form, which is called list method.

Mirror method: a method of representing the functional relationship between two variables with images, which is called mirror method.

Analytic method: the corresponding relationship of functions can be expressed by the analytic formula of independent variables, which is called analytic method.

3. Piecewise function

(1) Definition of piecewise function: In different parts of the domain, functions with different corresponding rules are called piecewise functions.

(2) Definition domain and value domain of piecewise function: the definition domain of piecewise function is the union of the definition domain of each segment, and its value domain is the union of the value domain of each segment.

4. The concept of mapping

If two sets A and B have a corresponding relationship F, and each element X and B in A always has a unique element Y, it is called a mapping from A to B. If the mapping from A to B satisfies that the images of different elements in A are different and each element in B has an original image, it is called a one-to-one mapping.

Third, the nature of the function:

Monotonicity of 1. function

(1) Definition: For any two independent variables x 1, x2, when X 1

(2) Features: The Y value of the increase (decrease) function increases (decreases) with the increase of the independent variable X value, that is, the image of increasing function rises from left to right, while the image of the decrease function decreases.

2. The parity of the function

(1) definition: for any x in the definition domain of function f(x), if f (-x) =-f(x) (f (-x) = f (x)), then f(x) is called odd function (even function).

(2) Symmetry of the definition domain: Symmetry of the function definition domain about the origin is a necessary condition for the function to be an odd (even) function.

(3) Symmetry of images: Is f(x) an odd (even) function? The image of f (x) is symmetrical about the origin (y axis).

3. The periodicity of the function

For the function f(x), if there is a non-zero constant t and f(x+t) = f (x) for any x in the definition field, then f (x) is called a periodic function, and the constant t is called the period of this function.

Fourth, power function, exponential function, logarithmic function

1. power function

The function y=za (constant a is a real number) is called a power function. Generally, we only consider the image and properties of power function when a= 1, 2,3,-1 and its simple application.

2. Exponential function

Power of (1) Fractional Exponent and Its Operational Properties

Definition: a =, a =, (a > 0, a ≠0, m, n ∈N* and >; 1);

Nature of operation: as? At=as+t, (as)t = ast, (ab)= asbs (where a >；; 0，b & gt0，s，t∈Q)。

(2) the definition of exponential function

The function y=a (constant a > 0 and a≠l) is called exponential function.

3. Logarithmic function

The definition and operational properties of (1) logarithm

① definition: if ab = n, Logan = b (a > 0, a≠ 1, n > 0).

② operational properties: loga (Mn) = logam+Logan, loga = logam-Logan, logaMn=nlogaM.

(M & gt0，N & gt0，a & gt0，a≠ 1)

③ identities: logal = 0, log.aa = 1, alogan = n (a > 0, a≠ 1, n > 0), etc.

(2) Logab = (b > 0，a > 0，c > 0: a ≠ 1.c ≠ 1)。

(3) let the function Y=logax (constant a > 0 and a≠ 1) be called logarithmic function.

4. Inverse function

The concept of (1) inverse function: Let the domain of function y =f(x) be A and the domain of value be C. If y is used from y=f(x),

X=φ(y), and for any one of C, there is a unique X corresponding to it in A, then x=φ(y) means that X is a function of Y, which is called the inverse function of the original function. Traditionally, the inverse function of y=f(x) is written as Y = f- 1 (x).

(2) Simple solution of the inverse function: first, express X with Y from y=f(x), and then exchange the positions of X and Y in the solution expression according to the habit (that is, take X as the independent variable and Y as the function value) to get the inverse function of the original function Y = f- 1 (x).

(3) The properties of reciprocal inverse function: The main properties of reciprocal inverse function are: ① the domain and value domain of reciprocal function, which are the domain and domain of the original function respectively; ② Two mutually inverse images are symmetrical about the straight line y=x; ③ The original function and its inverse function have the same monotonicity.

5. The relationship between exponential function and logarithmic function

Exponential function y = ax (a > 0, a≠ 1) and logarithmic function y = logax (a > 0, a≠ 1) are reciprocal functions, and their images are symmetrical about the straight line y=x, and their images and main properties are as follows.

Exponential function y = ax. (A > 0, a≠ 1) logarithmic function y = logax (A > 0, a≠ 1).

Field R (0, +∞)

Range (0, +∞) R

Image y

0 1

O x

y a> 1

O

0 0, r increases;

When 0

(2) when a > 1, it is added to (0, +∞);

When 0

Verb (abbreviation for verb) function image

1. Picture

There are two main ways to make an image of a function.

The first method is "tracing point method": take a list of values and trace points to connect lines.

To prevent the blindness of drawing, it can be divided into three steps: ① First, study the definition range and value range of the function to determine the range of the image; Secondly, the parity of the function is studied to determine the symmetry of the image; Finally, the monotonicity of the function is studied to determine the rising and falling trends of the image.

The second is the "transformation method", which uses the basic function image to draw through image transformation. There are three forms of image transformation.

Type:

Translation transformation: an image with y=f (x) and an image with y=f(x+h);

Let the image of y=f(x) be the image of y =-f (x)+k.

Symmetric transformation: transform an image with y=f(x) into an image with y =-f (x);

The image of y=f(x) is the image of y = f (-x);

The image with y=f(x) is the image with y =-f (-x);

The image with y=f(x) is the image with y = f- 1 (x) (the inverse function of the original function);

Let the image y=f(x) be the image y = f (2a-x) (if the function y=f(x) satisfies f(x-a) =f(a-x), then the image y=f(x) is symmetrical about the straight line y);

Combining the images with y=f(x) to make them symmetrical, and obtaining the image with y=f(|x|);

Take the image of y=f(x) and y = | f (x) |

Telescopic transformation: shorten (ω> 1) or extend (ω< 1) the abscissa of each point on the image with y=f(x) to the original abscissa, so that the image with y = f(ωx) can be obtained; The image of function y =Af(x) can be obtained by extending the ordinate of each point on the image of y = f (x) (a >1) (shortened when 0 < a <1) to the original a times.

First, the assembly part:

Example: 1, let a, then the following relationship is correct ()

A.B. C. D。

2. If the set is known, the set is ().

A.B. C. D。

3. Let the set m = {x ≤} and n = {x | x2+2x-3.

(A){ x | 0≤x < 1 }(B){ x | 0≤x < 2 }(C){ x | 0≤x≤ 1 }(D){ x | 0≤x≤2 }

4. Let A∩B= {y | y = x2, x ∈ r}, b = {y | y = 2-| x |, x ∈ r}, then a ∈ b =;

A∩？ U B= _。

5. As shown in figure 1- 1, the shaded part can be expressed as ().

(1)? U (A∩B) (B)？ u(A∠B)

(3)? u(A∪B)∩(A∪B)(D)(A∪B)∩？ U(A∩B)

Exercise 1. Let the set U = {1, 2,3,4,5}, A = {1, 2,3} and B = {2 2,5}, then A∩ (? U B)=()。

(A){2} (B){2，3} (C){3} (D){ 1，3}

2. If a = {x ∣ 2x+ 1 ∣ > 3} and b = {x ∣ x2+x-6 ≤ 0}, then a ∩ b = ().

(A)(-3，-2〖 1，+∞) (B)(-3，-2〗 1，2)

(C)∞-3，-2)∩( 1，2)〔D〕(-∞，-3〕∩( 1，2〕

3. Let A, B and U all be nonempty sets and satisfy a? b？ U, then the error in the following categories is ().

(1) (? U A)∪B=U (B)(？ u A)∨(？ U B)=U

(C)A∩(？ U B)=？ (D)(？ U A) ∩(？ U B)=？ U B

4. As shown in figure 1-2, U is a complete set, and M, P and S are three subsets of U, then the shaded part

The set represented by the branch is ().

(A)(M∩P)∩S(B)(M∩P)∩S

(C)(M∩P) ∩(？ America) (D)(M∩P)∩ (? United States)

5. if the complete set U = R, F (X) and G (X) are quadratic functions of x.

p={x∣f(x)& lt； 0}, q = {x ∣ g (x) ≥ 0}, then the solution set of inequality group {is used.

P and q are expressed as

6. Let a = {5, log2 (a+3)} and b = {a, b}. If a ∩ b = {2}, then a ∪ b =.

7. Given the set A = {A2, A+ 1, -3}, B = {A-3, 2A- 1, A2+ 1}, if A ∩ B = {-3}, find the value of the number A. 。

8. If a = {x ∣ y = LG (4x2-4)} and b = {y ∣ y = 2x2-3}, then a ∩ b =

9. If the solution set of inequality is, the value of.

Second, the concept and nature of function

Example 1. Which of the following functions represents the same function?

① f (x) = 1 and (x) = XO; ② f (x) = 1gx2，g(x)= 2 1gx；

③ f (x) = x2- 1 and g (x) = | x2-1| 4f (x) = ax2 (a ≠ 0) and g (t) = at2 (a ≠ 0).

2. Given the mapping f:(x, y) → (x+y, xy), find the image of (-2,3) under f and the original image of (2,3).

3. Find the domain of the following function.

( 1)y = 1g( 16-4x)+(x+ 1)0； (2)y =； (3)y =+LG cosx；

4. It is proved that f(x)= ex+ is a increasing function on (0, +∞).

5. The increasing interval of the function f (x) = log0.5 (x2-6x+8) is; The decreasing interval is.

6, in the interval (-∞, 0) for increasing function is ().

(A)y =-log 0.5( 1-x)(B)y =(C)y =-(x+ 1)2dy = 1+x2

7. Let A ∈ R and F (x) =-A be odd function, and find the value of a.

8. If f (x) = x2+asinx+x+8 and f (-2) = 10, then f (2) = _.

9. It is known that f(x) is an even function defined on (-1, 1) and a increasing function on the interval [0, 1]. If the inequality f (a-2)-f (3-a) < 0 holds, find the range of number A. 。

Exercise: 1 Among the following functions, the same as the function y = parity is ().

(A) y= |x+l|+|x- 1 | (B) y=2-x

(C) y= (D) y= wrong! The link is invalid. +

2. The function f(x) defined on R is the increasing function on (-∞, 2), and the symmetry axis y = f (x12) of the image is the straight line x=0, then the following relationship holds ().

f(- 1)& lt； f(3) (B)f(0)>f(3)(C)f(- 1)= f(-3)(D)f(2)& lt； Female (3)

3. The increasing interval of the function Y= log2 (-x2-2x+8) is.

4. If the function f(x) =8x2 +ax+5 is a increasing function on [1, +∞], then the range of a is.

5. The value range of x satisfying the inequality logx < 1 can be expressed as an interval.

6. If f(x)=, the value of f (-1)+f (-2)+f ()+f ()+f (-4)+f () is.

7. odd function f(x) decreases on R. For any real number x, the inequality f (kx)+f (-x2+x-2) >1; 0 is always true, and the range of k is realistic.

8. odd function knows that the function decreases at (0, +∞). It is proved that it decreases at (-∞, 0).

Third, power function, exponential function and logarithmic function

Example: 1, Calculate (1) to calculate the value of [125 +( +343);

(2) Find the value range of x satisfying the inequality x–1< x2.

3)

2. Find the domain of the following function:

( 1)、 ; (2)、

3. The range of known (1) is; (2) When, the value range is;

4. known. The value.

5. Study the domain, monotonicity and parity of functions.

Exercise 1. The solution set of equation 5x- 1x 10x = 8x is ().

(A){ 1，4| (B){ 1 }，(C) { }，(D){4，}

2. if 1 < x < y, the following inequality holds ().

(a) 3y-x < 3x-y (b) 3x-1< 31-y (c) 3x-1> 31-y, (d) logo.2 (x-l

4. The value range of the function y=2-x +6x- 17 is, and the increasing interval is.

5. It is known that log 18 9=a, 18b =5: A and b stand for log36 45.

6. The number of true propositions in the following propositions is ()

① ; ② ; ③ ; ④ is an exponential function; ⑤ is an exponential function;

7. Known, search (1); (2) ;

8, known ()

(1) is domain dependent; (2) judging parity and proving it; (3) Discuss monotonicity.

9, known, try to find the maximum and minimum value of the function.