The relationship between deterministic phenomena is often expressed as a functional relationship, that is, after the number of one phenomenon is determined, the number of another phenomenon is also completely determined, which is expressed as a strictFunctional relation。When one or several variables take a certain value, another variable has a certain value corresponding to it, then this relationship is called deterministicFunctional relation, recorded as y=f (x), where x is called independent variable and y is called dependent variable.[1]
For example, the relationship between the sales volume y and the sales volume x of a certain commodity can be expressed as y=px (p is the unit price);The relationship between the area S of a circle and the radius R can be expressed as S=π Rtwo；Enterprise's raw material consumption Y and output Xone. Consumption per unit output XtwoRaw material price XthreeThe relationship between them can be expressed as Y=XoneXtwoXthree；The relationship between the trading volume of a stock and the trading volume of the stock. When the trading price P remains unchanged, when the trading volume X of the stock is determined, its trading volume Y is also determined. The relationship between the three is: Y=PX.[1]
It is both a monomorphism from set A to set B and a epimorphism from set A to set B, so it is called a monomorphism from set A to set BBirefringence。[2]
Precautions
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Several notes on function definition:
(1) If
It is a function from set A to set B. Each element in set A must have an image and the image must be unique.
(2) If
It is a function from set A to set B. Then every element in set B may not have a primitive image, and when there is a primitive image, the primitive image may not be unique.
(3) According to the definition of function, we can know whether two functions are equal.You need to see whether the definition fields of the two are the same;For each element in the domain, whether its image under the action of these two functions is always the same.[2]
Establishment of functional relationship
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Method steps
For practical problems, it is very important to clarify various quantities and their relationships, and establish correct functional relationships.When establishing the functional relationship, first determine the independent variable and dependent variable in the problem, then list the equation according to their relationship to obtain the functional relationship, and then determine the function definition domain. When determining the definition domain, not only the analytic expression of the functional relationship, but also the meaning of the variable in the actual problem should be considered.[3]
Basic steps for establishing functional relationship:
The following example shows how to establish a functional relationship.
Example 1A mall sells 8000 pieces of certain goods at an original price of 70 yuan each. When the sales volume is less than 5000 pieces (including 5000 pieces), they will be sold at the original price. If the sales volume is more than 5000 pieces, they will be sold at a 20% discount.Try to establish a functional relationship between total sales revenue and sales volume.
Solution:Let the sales volume be x pieces, the total sales revenue be R yuan, and the functional relationship between the total sales revenue and the sales volume is
Example 2A factory produces a certain type of lathe, with an annual output of a set, which is produced in several batches. The production preparation fee for each batch is b yuan. If the products are evenly put into the market, and the next batch is produced immediately after the last batch is used up, that is, the average inventory is half of the batch. If the annual inventory fee for each set is c yuan, obviously, the inventory fee is high when the production batch is large;If the production batch is small, the number of batches will increase, soProduction preparation costIn order to select the optimal batch size, try to find out the relationship between the sum of inventory cost and production preparation cost in a year and the batch size.
Solution:Let the batch size be x, and the sum of inventory cost and production preparation cost is P (x). Since the annual output is a, the number of batches produced each year is
(Let it be an integer), then the production preparation cost is
, because the inventory is a, the inventory cost is
, so we can get
The definition field is (0, a], because x in this question is the number of lathes and batches
Is an integer, so x should only take the positive integer factor of a in (0, a].
Example 3A pasture will be built to cover an area of 100mtwoA row of 20m long rectangular walls is available. In order to save investment, one side of the rectangular wall is directly repaired with the old wall, and the other three sides are reconstructed as far as possible with the old walls removed. The insufficient part is newly built with new bricks purchased. It is known that it costs 24 yuan to renovate the 1m old wall, 100 yuan to renovate the 1m old wall, 200 yuan to construct the 1m new wall, and the reserved part of the old wall is represented by x,The whole investment is expressed in y, and y is expressed as a function of x.
Solution:The cost of the whole investment includes the cost of renovating the old wall, the cost of demolishing the old wall and renovating the new wall, and the cost of building a new wall, so the functional relationship is
Example 4The electricity price of a certain area last year was 0.8 yuan/(kW · h), and the annual electricity consumption was a kW · h. This year, the electricity price was reduced to between 0.55 yuan/(kW · h) and 0.75 yuan/(kW · h), while the expected electricity price of users was 0.4 yuan/(kW · h). Through calculation, the new electricity consumption after the price reduction is inversely proportional to the difference between the actual electricity price and the expected electricity price of users (the proportional coefficient is k), and the electricity cost in this area is 0.3 yuan/(kW · h),Write the functional relationship between the revenue y of the power sector and the actual electricity price x after the electricity price is lowered this year.[3]
Differentiation between functional relationship and correlation relationship
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Definition of correlation
When variable X takes a certain value, variable Y may take several values. These values show certain volatility, but always revolve around their average and follow certain laws.The uncertain quantitative relationship between variables is calledCorrelation。Features: The values of Y and X do not correspond one by one;The relationship between Y and X cannot be strictly expressed in functional formula, but there are rules to follow.
For example, the relationship between father's height Y and child's height X;The relationship between income level Y and education level X;Grain yield per mu Y and fertilization amount Xone, rainfall Xtwo, temperature XthreeRelationship between;The relationship between commodity consumption Y and resident income X;The relationship between commodity sales Y and advertising expenses X.[1]
The distinction between the two
The basis for distinguishing correlation and functional relationshipDetermination of dependent variable value: If the value of the dependent variable is determined and unique, the relationship between the two variables is calledFunctional relation；If the value of the dependent variable is uncertain, the relationship between the two variables is called correlation.
Example 5Try to determine whether the following variables belong to functional relationship or correlation relationship.
(1) Circle area and circle radius (2) Sales volume and sales volume of goods under price determination
(3) People's height and weight (4) Advertising expenses and sales
(5) Household monthly income and expenditure (6) Fertilization amount and yield per mu
(7) Education and Annual Income (8) Book Print Number and Book Price
(9) Commodity sales and commodity circulation expense ratio (10) Variable sales price and commodity sales
Solution:According to the definition and difference of functional relationship and correlation relationship, in this example, (1) and (2) are functional relationships, and the rest are correlation relationships.
be careful:The functional relationship and correlation between variables can be transformed into each other under certain conditions.When there is observation error, the functional relationship of variables with functional relationship is often expressed in the form of correlation.If we have a deep understanding of the relationship between variables with correlation and can incorporate all the factors that affect the change of dependent variables into the equation, then the correlation may also be transformed into a functional relationship.In addition, the correlation also has some regularity of change, so the correlation can often be approximately described in a certain functional form.The functional relationship of objective phenomena can be studied by means of mathematical analysis, while the correlation of objective phenomena must be studied by means of correlation and regression analysis in statistics.[1]
