Functional relationship

Definition

Sets A and B are two given sets,

is a binary relationship from the set A to the collection B. If this binary relationship is also satisfied with the nature: for each element

, there is a unique element

, makes binary elapses

This binary relationship is a function or (B> "mapping from the collection A to the collection. Record

or

can also be rewritten as

, Where Y is the like , and X is called Y. As a result of the collection A is the defined domain , all elements obtained under the function

Icon or value domain value domain

.

gives a definition of the subdivision to further distinguish between different characteristics.

Set

is a function from the collection A to the collection B.

(1) If

, the function

is a single shot from the set A to the collection B.

(2) If

, the function

is a full shot from the set A to the collection B.

(3) If the function

is both a single shot from the set A to the collections B. It is also a full shot from the collection A to the collection B, which is called a double shot from the set A to the collection B.

Notes

About function definitions Description:

(1) If

is from Collection A to Collection B A function, then each element in the collection A must be like it, and the like must be unique.

(2) If

is a function from the collection A to the collection B. Then, each element in the collection b does not necessarily have an originality, and when there is an idea, it is not necessarily unique.

(3) It is understood according to the definition of the function, whether the two functions are equal. Need to see: The definition domain of the two is the same; for each element in the domain. It is the same as the image under these two functions.

Function relationship

method step

For actual problems, it is important to establish the relationship between these amounts and quantities, and establish the correct functional relationship is very important. . When establishing a functional relationship, you must first determine the self-variable and due to variables in the problem, and then list the equality according to the relationship between them, and determine the functional relationship, then determine the function definition domain, determine the definition domain, not only to consider To the parsing of the functional relationship, considering the meaning of variables in practical problems.

Establishing a functional relationship:

1 Clear problems due to variables and arguments, and represent appropriate markers;

2 find equal relations To establish a functional relationship;

3 determination of the defined domain of the function.

Examples

The following example shows how to establish a functional relationship.

Example 1 A shopping mall sells 8,000 pieces of items, each original price of 70 yuan, when the sales volume is within 5,000 pieces (including 5000 pieces), according to the original price, exceeding 5000 parts, playing high sales. Try to establish a functional relationship between total sales revenue and sales volume.

Solution: The sales volume is X pieces, the total sales revenue is R yuan, the function relationship between the total sales income and the sales volume is

Example 2 A plant produces some model of lathe, annual output is a station, subsequent batch production, each batch of production The preparation fee is b yuan, and the product is evenly put into the market. High inventory fees; the production batch is increased, and the production preparation fee is high. In order to choose the optimal batch, try the relationship between inventory fees and production preparation fees in the year.

Solution: Set the batch of x, inventory fees and production preparation fee and P (x), due to annual production is A, so the batch produced annually is

(set to an integer), the production preparation fee is

, due to the inventory A, so the inventory fee is

, so you can get

Defining the domain (0, a], due to the number of the lathes in this question, the number of batch

is an integer, so X only (0, A ] A positive integer factor in A.

Example 3 a ranch to build a rectangular wall covering an area of ​​100m ² , a row of 20m long The old wall is available for use, in order to save investment, the rectangular wall is directly repaired with old walls, and the three-sided old wall is used to rebuild the old wall, and the new bricks purchased by the shortage are new. It is known to renovate 1M old wall. 24 yuan. , Demolished 1M old wall reform to build 1M new wall to take 100 yuan, build 1M new wall to need 200 yuan, and the part reserved by the old wall is indicated by x, and the entire investment is used, and Y is a function of x.

Solution: The cost of the entire investment includes the cost of renovating the old wall, the cost of removing the old reform, and the cost of building a new wall, so the functional relationship is

Example 4 The annual electricity price in a certain area is 0.8 yuan / (kW · h), the annual electricity consumption is a kw · h, this year will The electricity price is reduced to 0.55 yuan / (kW · h) to 0.75 yuan / (kW · h), and the user expects the electricity price of 0.4 yuan / (kW · h), the measurement, the new electricity consumption after the electricity price is lowered. The actual electricity price and the user expectation of the electricity price (K), the power cost of the region is 0.3 yuan / (kW · h), and the product price of the electricity department and the actual price of the actual electricity price X are written out in this year. Relationship.

solution: Revenue = actual electricity amount × (actual price - cost price).

So the functional relationship is

Several common function relationship model

One function model

(

is constant,

).

Inverse proportional function model

is constant,

).

Quadruple function model

(

is constant,

).

Index type function model

(

is constant,

,

).

logarithmic function model

(

is constant,

,

).

Power Function Model

is constant,

).

< H2> Function relationship and related relationship Distinguishing

Related Relationship

When the variable X takes a certain value, the value of the variable Y may have several, these numerical values A certain volatility, but always surrounds their average and follow certain regular changes. This uncertain quantity relationship between variables is called related relationships . Features: Y and X correspond to the value; Y and X relationship cannot be used in strict expression, but there are regularities.

For example: the relationship between the father's height Y and the child's height x; the relationship between the income level Y and the degree of education of the education; Yence Yield Y and Fertilization amount x ₁ Relationship between rainfall x ₂ , temperature x ₃ ; the relationship between the consumption of the goods and the income X of the residents; the product sales Y and the advertising fee The relationship between X.

Differences of the two

Differentiated relationships and functional relationships are based on Deterministic value of variable value : value due to variables It is determined, unique, the relationship between the two variables is called the function relationship ; if it is uncertain due to the value of the variable, the relationship between the two variables is called a related relationship. .

Example 5 Tests Determine the relationship between the following variables, or related relationships.

(1) Circular area and round half (2) Price determination of sales and sales volume of the goods

(3) people's height and weight (4) Commodity advertising fee expenditure With sales

(5) Family monthly income with monthly expenditure (6) Fertilizer and mu yield

(7) Culture and annual income (8) Book print and book Price

(9) Product sales and commodity circulation cost rate (10) variable sales price and product sales

solution: According to the functional relationship and Definition and difference of related relationships, in this example, items (1), (2) are functional relationships, and the rest are related relationships.

Note: The function relationship between the variables and related relationships can be transformed with each other under certain conditions. The variable of the functional relationship is the functional relationship thereof often exhibit in the relevant form when there is observation error. The connection between the variables with the correlation relationship can be incorporated with a profound regular understanding, and it is possible to convert all the factors that affect variables into equations. In addition, the relevant relationship also has a certain change rule, so the associated relationship can often be approximated in the form of a certain function. The functional relationship of objective phenomena can be studied by methods of mathematical analysis, and the relationship between objective phenomena must be studied, and the relevant and regression analysis methods must be used in statistics.