1. mapping: the concept of A B When understanding the concept of mapping, it should be noted that the elements in (1)A must be unique and unique; (2) The elements in B may not all have original images, but the original images are not necessarily unique. For example, if (1) is a mapping from a set to, the following statements are correct: A, every element in the set must have an image B, every element in the set must have an original image C, and the original image of every element in the set is a unique D, which is a set of images of the elements in the set (A); (2) If the image of a point under the action of mapping is _ _ _ _ _ _ _ (a: (2,-1)); (3) If,,, is mapped to, is mapped to, and has a function to (A: 8 1, 64,81); (4) Set a set, and the mapping satisfies the condition that "any mapping is odd", and there are _ _ _ _ such mappings (a:12); (5) Let it be the mapping from set A to set B. If B={ 1, 2}, it must be _ _ _ _ (A: or {1}).

2. Function: A B is a special mapping. Specially defined field A and value field B are non-empty number sets! Accordingly, the function image has at most one common point with the vertical line of the axis, but there may be no common point with the vertical line of the axis, or there may be any point. If the function is known at (1), the number of elements contained in the set is (A: 0 or1); (2) If the function's domain and value domain are closed intervals, then = (A: 2)

3. The concept of the same function. The three elements that make up a function are definition domain, value domain and corresponding rules. The scope can be uniquely determined by the domain and the corresponding rules, so when the domain and the corresponding rules of two functions are the same, they must be the same function. If a series of functions have the same analytical formula and the same range, but their definition ranges are different, they are called "tianyi functions", so there are _ _ _ _ * tianyi functions, and the analytical formula and range are {4, 1} (A: 9).

4. Common methods for finding function domain (the principle of domain priority should be established when studying function problems):

(1) According to the requirements of analytical formula, such as the roots of even roots are greater than zero, the denominator cannot be zero, the logarithm is the median, and the triangle has the maximum angle and the minimum angle. For example, the domain of the (1) function is _ _ (a:); (2) If the domain of the function is r, then _ _ _ _ _ _ _ (a:); (3) If the domain of the function is 0, the domain of the function is _ _ _ _ _ _ _ (a:); (4) Set a function: ① If the domain of is r, it is the range of real numbers; ② If the range of is r, then the range of real numbers (a: ①; ② )

(2) Determine the range of independent variables according to the requirements of practical problems.

(3) Definition domain of composite function: If the definition domain is known as, the definition domain of composite function can be solved by inequality; If the domain is known as, then the domain of is equivalent to the domain of (that is, the domain of). For example, (1) If the domain of a function is, the domain of is _ _ _ _ _ _ _ (a:); (2) If the domain of the function is, the domain of the function is _ _ _ _ _ _ _ _ (a:).

5. The method of finding the function value domain (maximum value):

(1) collocation method-quadratic function (quadratic function has two kinds of maxima in a given interval: one is to find the maximum in a closed interval; The second is to find the maximum value of interval (motion) and symmetry axis (motion). Find the maximum value of quadratic function, don't forget the combination of numbers and shapes, pay attention to "two looks": look at the opening direction; Second, look at the relative position relationship between the axis of symmetry and a given interval), such as (1) to find the range of function (a:); (2) When the function gets the maximum value at, the value range of is _ _ (a:); (3) If the known image intersects with (2, 1), the range of is _ _ _ _ _ (A:).

② The image of a function and the image of its inverse function are symmetrical about a straight line. Note that the image of this function is the same as that of another function. If (1) knows that the image of the function passes through the point (1, 1), then the image of the inverse function must pass through the point _ _ _ (a: (1,3)); (2) Given a function, if the function and the image are symmetrical about a straight line, the required value (a:);

③ 。 If the function is known to be (1), the solution of the equation is _ _ _ _ (a:1); (2) Let the image of function f(x) be symmetric about point (1, 2) and have an inverse function, and f (4) = 0, then = (A:-2).

④ Two functions that are reciprocal functions have the same monotonicity and odd function. If it is known that it is a increasing function on a graph and the point on its image is its inverse function, then the solution set of the inequality is _ _ _ _ _ _ _ (a: (2,8));

⑤ If the definition domain is A and the value domain is B, then there is,

, but.

9. Functional equivalence.

(1) Characteristics of the domain of a function with parity: the domain must be symmetrical about the origin! Therefore, when determining the parity of a function, we must first determine whether the function domain is symmetrical about the origin. If this function,

Is an odd function, where the value is (a: 0);

(2) Common methods to determine the parity of a function (if the analytical formula of a given function is complicated, it should be simplified before judging its parity):

① Definition method: such as judging the parity of a function (a: odd function).

② Equivalent form of the definition of function parity: or (). Such as the parity of judgment. (A: Even function)

③ Mirror image method: the mirror image of odd function is symmetrical about the origin; The image of an even function is symmetrical about the axis.

(3) The essence of function parity:

(1) If odd function has monotonicity in the interval symmetrical about the origin, its monotonicity is exactly the same; Even functions are monotonic if they are symmetric about the origin.

(2) If odd function has an inverse function, then its inverse function must still be odd function.

③ If it is an even function, then. If the even function defined on R is a decreasing function and =2, then the solution set of inequality is _ _ _ _ _. (a:)

(4) If the odd function domain contains 0, it must exist, so it is neither sufficient nor necessary for odd function. In case of odd function, real number = _ _ _ (A: 1).

⑤ Any function defined on a symmetrical interval about the origin can be expressed as "the sum (or difference) of a odd function and an even function". Let it be any function whose domain is r,,. (1) Judge the parity of sum; ② If the function is expressed as the sum of odd function and even function, then = _ _ _ (A: ① is even function and odd function; ② = )

⑥ The parity characteristic of composite function is: "Even if it is inside, it is odd if it is outside".

⑦ There are infinite odd-even functions (and the domain is any set of numbers that are symmetrical about the origin).

10. Monotonicity of the function.

(1) Common methods for judging monotonicity or monotone interval of functions:

① Common methods for solving problems: definition method (determination of value-difference-deformation-number) and derivation method (in the interval, if there is always one, it is increasing function; On the other hand, if it is an increasing function within an interval, please pay attention to the difference between them. If it is known that the function is increasing function in the interval, the value range of is _ _ (a:);

(2) In the choice of fill-in-the-blank questions, you can also use the combination of numbers and shapes, special value method, etc., and pay special attention to it.

The application of visualization and monotonicity of type I function in solving problems: the increasing interval is 0 and the decreasing interval is 0. For example, (1) If the function is a decreasing function (-∞, 4) in the interval, then the range of real numbers is _ _ _ _ (a:); (2) The given function is increasing function in the interval, and the value range of real number is _ _ _ _ (a:); (3) If the range of function is r, the range of real number is _ _ _ _ (a: and);

③ Composite function method: the monotonicity of composite function shows the same increase but different decrease, for example, the monotonic increasing interval of the function is _ _ _ _ _ (a: (1,2)).

(2) Special reminder: When looking for monotonous intervals, first, don't forget the definition domain. If the function is a subtraction function in the interval, it is the range of the value to be searched (a:); Second, symbols ""and ""or ""shall not be added between multiple monotonous intervals; Thirdly, monotone intervals should be expressed by intervals, not by sets or inequalities.

(3) Have you noticed the reversal of monotonicity and parity of functions? (1) comparison size; (2) solving inequalities; (3) Find the parameter range). If it is known that odd function is a subtraction function defined on a real number field, if it is. (a:)

1 1. Normal image transformation

① Shift the image of the function by one unit to the left along the axis to get the image of the function. If the image and the image are symmetrical about a straight line, and the image is moved to the right by 1 unit to get the image, it is _ _ _ _ _ _ _ _ _ (A:).

② The image of the function (obtained by translating the image of the function to the right along the axis). If (1), the minimum value of the function is _ _ (a: 2); (2) To get an image, you only need to make an image about _ _ _ axis symmetry, and then translate 3 units to _ _ _ (a:; Right); (3) The number of intersections between the image and the axis of the function is _ _ _ _ (A: 2)

③ The image of function+is obtained by translating the auxiliary image of function along the axis;

(4) The image of the function+is obtained by translating the auxiliary image of the function along the axis unit; If the image of the function moves 2 units to the right, and then moves 2 units down, if the obtained image is symmetrical with the original image about a straight line, then (A: C)

⑤ Stretch the image of the function to the original image along the axis to get the image of the function. For example, (1) change the abscissa of all points on the function image to the original image (the ordinate is unchanged), and then shift the image to the left by 2 units along the axis direction, and the corresponding function of the obtained image is _ _ _ _ _ (a:); (2) If the function is an even function, the symmetric equation of the function is _ _ _ _ _ _ (A:).

⑥ The image of the function is obtained by scaling the image of the function to the original multiple along the axis.

Symmetry of functions.

① The image of the function satisfying the condition is symmetrical about a straight line. If it is known that the quadratic function satisfies the condition and the equation has equal roots, then = _ _ _ (a:);

(2) The symmetry point of a point about the axis is; The symmetric curve equation of the function about the axis is:

(3) The symmetry point of a point about the axis is; The symmetric curve equation of the function about the axis is:

(4) The symmetrical point of the point about the origin is; The symmetric curve equation of the function about the origin is:

(5) The symmetrical point of the point about the straight line is; The equation for the symmetric curve of a curve about a straight line is. In particular, the symmetry point of a point about a straight line is; The equation for the symmetric curve of a curve about a straight line is

; The symmetry point of a point about a straight line is; The equation for the symmetric curve of a curve about a straight line is. The known function, if the image is, is symmetrical about the straight line and the origin, and the corresponding resolution function is _ _ _ _ _ _ _ (a:);

⑥ The equation of curve symmetry curve about this point is. If the image of the function sum is symmetric about the point (-2,3), then = _ _ _ _ _ (A:)

⑦ The image is a hyperbola, and its two asymptotes are straight lines respectively.

(determined by the denominator being zero) and a straight line (determined by the coefficients in the numerator and denominator), and the center of symmetry is a point. If it is known that the function image is symmetric about a straight line and symmetric about a point (2, -3), then the value of a is _ _ _ _ _ _ _ (a: 2).

(8) Keep the original image above the axis, make a symmetrical figure about the axis for the image below the axis, and then erase the image below the axis to obtain the image; First, keep the image on the right side of the axis, erase the image on the left side of the axis, and then make the image on the right side of the axis into a symmetrical figure. For example, (1) makes an image of the sum of functions; (2) If the function is a odd function defined on R, the image of the function is symmetric about _ _ _ _ (A: axis).

Reminder: (1) From the conclusion 23456, it can be seen that the problem of finding symmetric curve equation is essentially transformed into a symmetric problem of finding points with method of substitution; (2) Prove the symmetry of the function image, that is, prove that any point in the image is still on the image relative to the symmetry center (symmetry axis); (3) There are two aspects to prove the symmetry of an image: ① to prove that the symmetry point of any point in the world about the symmetry center (symmetry axis) is still on the top; ② Prove that the symmetry point of any point on the plane is still on the plane. Such as (1) known function. Verification: the image of the function is a central symmetric figure about the point; (2) Let the equation of curve c be that the curve is obtained by moving c parallel to the unit length along the axis and the positive direction of the axis respectively. ① Write the equation of curve (a:); ② Prove that curve C is symmetrical about this point.

13. The periodicity of the function.

(1) analogy "trigonometric function image":

(1) If the image has two symmetry axes, it must be a periodic function, and one period is;

② If the image has two symmetrical centers, it is a periodic function, and one period is;

③ If the image of a function has a symmetrical center and axis, then the function must be a periodic function with a period of;

If it is known that the function defined on is a odd function with a period of 2, then the equation has at least _ _ _ _ _ _ real roots (A: 5).

(2) From the definition of periodic function "If the function is satisfied, it is a periodic function with a period of":

① If the function is satisfied, it is a periodic function with a period of 2;

(2) If the constant holds, then;

(3) if constant is established, then.

Suppose (1) is Upper odd function, then, if, then, it is equal to _ _ _ _ (a:); (2) The even function defined in satisfaction is a decreasing function in the world. If it is the two internal angles of an acute triangle, the size relationship is _ _ _ _ _ _ _ (a:); (3) Known as an even function, and =993, = odd function, evaluated (A: 993); (4) Let it be a function whose domain is R, then = (A:)

14. Exponential type and logarithmic type:

, ,, , , , , , , , , 。 For example, the value of (1) is _ _ _ _ _ _ (a: 8); The value of (2) is _ _ _ _ _ _ _ _(A:)

15. Comparison between exponent and logarithm: (1) Using monotonicity of function with the same base; (2) Business or business law; (3) Use intermediate quantity (0 or1); (4) Image contrast after transforming the same index (or the same true number).

16. The application of the function. (1) The general steps to solve mathematical application problems are as follows: ① Examining the problems-reading the problems carefully, accurately understanding the meaning of the problems, clarifying the actual background of the problems, and finding the memory relationship between quantities; (2) Modeling-transforming practical problems into corresponding mathematical problems through abstract generalization, and at the same time, don't forget to mark the definition fields that conform to practical significance; ③ Solving the model-solving the obtained mathematical problem; (4) Regression —— Regression of the mathematical results to the actual problems. (2) Common function models are as follows: ① Establish a linear function or a quadratic function model; ② Establish a piecewise function model; ③ Establish an exponential function model; ④ Preparation type.

17. Abstract function: Abstract function usually refers to a function problem that gives only some other conditions (such as definition domain, monotonicity, parity, analytic recursion, etc.) without giving the specific analytical expression of the function. ). Common methods to solve abstract function problems are:

(1) Use the model function for analogy. Several common abstract functions:

① Proportional function type:-;

② Power function type:-;

③ Exponential function type:-;

④ Logarithmic function type:-;

⑤ trigonometric function type:-. If it is known that it is a odd function defined on R and a periodic function, and if its minimum positive period is t, then _ _ _ _ (A: 0).

(2) Using the properties of functions (such as parity, monotonicity, periodicity, symmetry, etc.). ) Deductive exploration: For example, (1) let a function represent that the remainder is divided by 3, then any one has a, b, c, d and (a); (2) Let it be a function defined on the set of real numbers R, and satisfy, if, (a:1); (3) Let it be the odd function defined on the graph, and prove that the straight line is the symmetry axis of the function image; (4) Functions whose domain is known to satisfy and monotonically increase with time. If and, the sign of the value is _ _ _ _ (A: negative number).

(3) Use some methods (such as assignment method (order = 0 or 1, find or, order or etc.). ), recursive method, reduction to absurdity, etc. ) to explore logic. If (1) if, satisfying

, then the parity is _ _ _ _ _ (A: odd function); (2) If, meet

Then the parity of is _ _ _ _ _ (A: even function); (3) It is known that odd function is defined on the top. When the image is as shown on the right, the solution set of inequality is _ _ _ _ _ _ _ _ _ (a:); (4) Let the domain be, for any, all, and, and, and, ① prove that it is a decreasing function; ② Solve inequalities. (a:).

Function value definition field training problem

1. Given the domain of function g(x)=f(3-2x), the domain of function f(x) is _ _ _ _ _.

2.y = 1/[(x 2+2x+6) 0.5] Let x 2+2x+6 be t and (x 2+2x+6) 0.5 be a.

3. The domain is the range of the independent variable X in the function y=f(x).

4. If x, z and y are all positive numbers, x+y+z= 1, find the minimum value16/x3+81/8y3+1/27z3.

5. Find the value of a so that f(x) is a monotone function.

6. A circular fountain should be built in the park, and a cylinder OA should be installed in the center of the fountain perpendicular to the water surface, where O is right in the center of the circular water surface, OA =1.25m. The nozzle placed at the top of the cylinder A sprays water outwards, and the water flows down along a parabolic path with the same shape in all directions, and the parabolic path on any plane passing through OA is as shown in the figure. In order to make the water flow more beautiful, the designed water flow reaches the maximum height of 2.25m from the water surface at the distance from OA1m.. If other factors are not considered, how many meters should the radius of the pool be at least to prevent the spouted water from falling out of the pool?

7. Design a poster with an area of 4840 square centimeters and an aspect ratio λ (λ

Leave 8 cm blank at the top and bottom of the picture and 5 cm blank at the left and right sides. How to determine the height and width of the picture to minimize the poster paper area? If so, what is the value of λ to minimize the paper used in the poster?

8. the distance between a and b is s kilometers, so the car should drive from a to b at a constant speed, and the speed should not exceed c kilometers/hour.

At that time, it was known that the transportation cost of a car per hour (in yuan) consisted of variable part and fixed part. The variable part is directly proportional to the square of velocity v (km/h), and the proportional coefficient is b, and the fixed part.

9. If the function f (x- 1) = x2-2x+3, then f (x) = _ _ _ _ _ _ _ _ _, and f (x+1) = _ _ _ _ _.