The history of function concept is 1. The early concept of function-the function under the geometric concept (G Galileo, meaning, 1564- 1642) in the book Two New Sciences, almost all contain the concept of the relationship between functions or variables, and the relationship between functions is expressed in the language of 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. In 2008, Bernoulli Johann (Switzerland, 1667- 1748) defined the concept of function on the basis of the concept of Leibniz function: "a quantity consisting of 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 0/9th century-the function under corresponding relation 182 1 years ago, Cauchy (France, 1789- 1857) gave a 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 function with the fuzzy concept of "ordered couple" in the Outline of Set Theory, avoiding the two 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 x >; 0, the image passes through three or one quadrant 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. )