Why do you want to learn functions? Simply put, if you ask this question, the answers may be varied, hehe. When did the function appear? Modern mathematics began to study functions. The appearance of function is a great progress compared with the era without function. It represents the difference between thinking mode and thinking angle, and it is the arrival of a new mathematical era. Function is a powerful mathematical tool to solve problems. As a basic subject, mathematics has penetrated into almost all social and natural disciplines, and the influence of functions can be seen from this. The following is the development history of the concept of material function copied from Baidu Encyclopedia 1. The early concept of function-function under geometric concept (G Galileo, meaning, 1564- 1642) in the book Two New Sciences, almost all contain the concept of function or variable relationship, with words and proportions. Descartes (France, 1596- 1650) noticed the dependence of one variable on another around his analytic geometry 1673. However, because he didn't realize that the concept of function needed to be refined at that time, no one had defined the function until Newton and Leibniz established calculus in the late17th century. 1673, Leibniz first used "function" to express "power". Later, he used this word to represent the geometric quantities of each point on the curve, such as abscissa, ordinate, tangent length and so on. At the same time, Newton used "flow" to express the relationship between variables in the discussion of calculus. 2./kloc-function concept in the 8th century-function under algebraic concept 17 18 johann bernoulli (Swiss, 1667- 1748) defined the concept of function on the basis of the concept of Leibniz function: "From any variable and any form of constant, He means that any formula composed of variable X and constant is called a function of X, and he emphasizes that functions should be expressed by formulas. 1755, Euler (L. Euler, Switzerland, 1707- 1783) defines a function as "if some variables depend on other variables in some way, that is, when the latter variable changes, the former variable also changes, and we call the former variable a function of the latter variable." Euler (L. Euler, Switzerland, 1707- 1783) gave a definition: "The function of a variable is an analytical expression composed of this variable and some numbers or constants in any way." He called the function definition given by johann bernoulli's analytic function, and further divided it into algebraic function and transcendental function, which was considered as "arbitrary function". It is not difficult to see that Euler's definition of function is more universal and extensive than johann bernoulli's. 3./kloc-the concept of function in the 9th century-the function under corresponding relation 182 1 year, Cauchy (France, 1789- 1857) gave the definition from the definition of variables: "Some variables have certain relationships. The word independent variable appeared for the first time in Cauchy's definition, and pointed out that functions don't need analytic expressions. However, he still believes that functional relationships can be expressed by multiple analytical expressions, which is a great limitation. 1822, Fourier (France,1768-1830) found that some functions have also been expressed by curves, or they can be expressed by one formula, or they can be expressed by multiple formulas, thus ending the debate on whether the concept of functions is expressed by only one formula and pushing the understanding of functions to a new level. In 1837, Dirichlet (Germany, 1805- 1859) broke through this limitation and thought that it was irrelevant how to establish the relationship between x and y. He broadened the concept of function and pointed out: "For every definite value of X in a certain interval, Y has one or more definite values. This definition avoids the description of dependence in function definition and is accepted by all mathematicians in a clear way. This is what people often call the classic function definition. After the set theory founded by Cantor (German, 1845- 19 18) played an important role in mathematics, veblen (American, veblen, 1880- 1960) used "set" and ". 4. The concept of modern function-function under set theory1914 F. Hausdorf defined the function with the fuzzy concept of "ordered couple" in the Outline of Set Theory, avoiding the fuzzy concepts of "variable" and "correspondence". In 192 1, Kuratowski defined "ordered pair" with the concept of set, which made Hausdorff's definition very strict. In 1930, the new modern function is defined as "If there is always an element Y determined by set N corresponding to any element X of set M, then a function is defined on set M, and it is denoted as y=f(x). Element x is called an independent variable and element y is called a dependent variable. " The terms function, mapping, correspondence and transformation usually have the same meaning. But the function only represents the correspondence between numbers, and the mapping can also represent the correspondence between points and between graphs. It can be said that the mapping contains functions. Proportional function: the image of the proportional function y=kx(k is constant, k≠0) is a straight line passing through the origin. When k >: 0, the image passes through three quadrants and rises from left to right, that is, y increases with the increase of x; When k < 0, the image passes through two or four quadrants and descends from left to right, that is, y decreases with the increase of X. It is precisely because the image of the proportional function y=kx(k is constant, k≠0) is a straight line that we can call it a straight line y=kx. (In addition, the Chinese name "function" comes from China mathematician Li (1868). As for why this concept is translated in this way, the book explains that "whoever believes in this variable is the function of that variable"; "Faith" here means tolerance. ) In-depth study of a function When studying a function, Xu Ruohan should further study its practical application and how to change the image position according to the requirements of middle school. First, the piecewise function in practical problems [example 1] (Wuhan, 2005) Xiaoming rode his bike from home to school in the morning, going uphill first and then downhill. The itinerary is as shown. How long will it take Xiao Ming to ride home from school if the speed of going up and down the hill remains the same when he returns? Analysis: the speed of uphill and downhill is different, so the problem should be studied in two sections. According to the information provided by the function image, we can know that when Xiaoming goes to school from home, the uphill distance is 3600 meters and the downhill distance is 9600-3600 = 6000 meters. ∴ uphill speed is 3600÷ 18=200 (m/min), and downhill speed is 6000 ÷ (30- 18) = 500 (m/min). Xiao Ming goes home, the uphill distance is 6000m, and the downhill distance is 3600m m. Second, the application in physics [Example 2] (Huanggang City, 2004) When a class of students explored the relationship between spring length and external force, the corresponding data obtained from the experimental records were as follows: Find the resolution function of Y about X and the range of independent variables. Analysis: According to the knowledge of physics, the spring is deformed (elongated) under the action of external force (gravity of hanging heavy objects), and the relationship between external force and pointer position can be expressed by a linear function; But the external force on each spring is limited, so we must find the range of independent variables. According to the known data, let y=7.5 and x=275 ∴ function be that the dividing point of two segments is x=275, not x=300. Third, the application of linear translation [Example 3] (Heilongjiang Province in 2005) In the rectangular coordinate system, points A (-9,0), P (0 0,3) and C (0, 12) are known. Q: Is there a point Q on the X-axis, so that the quadrilateral with points A, C, P and Q as its vertices is a trapezoid? If it exists, find the analytical formula of straight line PQ; If it does not exist, please explain why. Analysis: Which two sides are parallel in the studied trapezoid? There are two possibilities: if, that is, the straight line CA is translated, the analytical expression of the straight line CA can be obtained after the point P is translated. If the straight line PA is translated, the analytical expression of the straight line: How to understand the concept of function when the straight line intersects the X axis at the point (-36,0). Cao Yang function is an extremely important basic concept in mathematics. In middle school mathematics, functions and their related contents are rich and heavy. Looking back on the development history of the concept of function, Leibniz first adopted "function" as a mathematical term. He first put forward the concept of function in his paper 1692, but its meaning is quite different from the current understanding of function. In modern junior high school mathematics curriculum, the definition of function is "variable theory". That is, in a certain change process, there are two variables X and Y. If there is a unique definite value corresponding to each definite value of X within a certain range, then Y is called a function of X, X is called an independent variable and Y is called a dependent variable. It clearly points out that the independent variable X can take any value within a given range, and the dependent variable Y takes a unique and certain value every time according to certain laws. But junior high school does not require mastering the range of independent variables (just look at several functions to be learned in junior high school, and you will know that this definition is completely sufficient and easy to understand for junior high school students). The concept of function is very abstract, which is difficult for students to understand. To understand the concept of function, we must clarify two points: First, we must clarify the relationship between independent variables and dependent variables. In a certain change process, there are two variables X and Y. If Y changes with X, then X is called independent variable and Y is called dependent variable. If x changes with y, then y is called independent variable and x is called dependent variable. Second, the core of function definition is "one-to-one correspondence", that is, given the value of an independent variable X, there is a uniquely determined value of a dependent variable Y corresponding to it. Such correspondence can be "one independent variable corresponds to one dependent variable" (abbreviated as "one-to-one") or "several independent variables correspond to one dependent variable" (abbreviated as "many-to-one"), but it cannot be "one independent variable corresponds to multiple dependent variables". The following figure 1 illustrates the corresponding relationship (where x is the independent variable and y is the dependent variable): whether "one-to-one", "many-to-one" and "one-to-many" are functions. The following figure 1 gives four examples to help you understand the concept of function: Example 1 The length of a spring is 10cm, when the spring is f (. The length of the spring is expressed by Y. The measured data are as follows: Table 1: Table 1 tension F(kg) 1 2 3 4 … spring length y(c) … Is spring length y a function of tension f? Analysis: Information can be read from the table. When the tensile forces are 1kg, 2kg, 3kg and 4kg respectively, they all correspond to the length y of a spring, which satisfies the definition of the function, so the length y of the spring is a function of the tensile force f ... Usually, the first line of the function given in tabular form is the value of the independent variable, and the second line is the value of the dependent variable. Example 2 Figure 2 shows the highest and lowest temperatures in a certain area every month of the year. Figure 2 Figure 2 describes the relationship between which variables? Can you regard one of the variables as a function of the other? Analysis: Three variables are given in the figure, namely the highest temperature, the lowest temperature and the month. As can be seen from the figure, the maximum and minimum temperatures change with the change of the month, and the maximum and minimum temperatures of each month are unique, so the maximum (or minimum) temperature is a function of the month. We can also find that the highest temperature in July and August is the same, which means that two independent variables correspond to the same dependent variable. Generally speaking, the horizontal axis represents the independent variable and the vertical axis represents the dependent variable for functions given in the form of images. Example 3 Is the relationship between the following variables a functional relationship? Explain why. (1) the relationship between the area s and the radius r of a circle; (2) When the car is traveling at a speed of 70km/h, the relationship between the distance traveled by the car s (km) and the time spent t (hours); (3) The area of an isosceles triangle is the relationship between its base length y (cm) and its height x (cm). Analysis: (1) The relationship between the area s and the radius r of a circle is that when the radius is determined, the area s of the circle is also uniquely determined, so the relationship between the area s and the radius r of the circle is a functional relationship. (2) The relationship between the distance S (km) and the time t (hour) used is that when the time t is determined, the distance s is also uniquely determined, so the relationship between the distance S (km) and the time t (hour) used is functional. (3) The relationship between the bottom length ycm and the top height xcm is that when the top height X is determined, the bottom length Y is also uniquely determined, so the relationship between the bottom length ycm and the top height xcm is a functional relationship. Generally speaking, the function given in the form of relation has dependent variables on the left of the equal sign and unknowns on the right of the equal sign as independent variables. Example 4 In the following images, what can't express the functional relationship is () analysis: In the above four images, A, C and D can all express the functional relationship, because any given value of the independent variable X has a unique Y value corresponding to it, but in Figure B, any given value of the independent variable X has two different Y values corresponding to it, so this question should choose 2.9] Let M be four digits less than 2006, and it is known that there is a positive integer N. 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