The concept of set and function 1.2 function and its representation.

Objective: To correctly understand the concept of function, describe the function with sets and corresponding languages, and understand the role of correspondence in describing the concept of function.

2. Understand the three elements of the function through a large number of examples;

Master the method of judging whether two functions are equal;

4. The activity of abstracting and generalizing the concept of function from practical problems.

Main cognition: the concept of function and the three elements of function.

Process:

We study the function, which was first adopted by the German mathematician Leibniz. Later, Van Buren and Linner revealed the essence of the concept of function with the viewpoint of set and correspondence. When translating algebra in China, mathematician Li was asked to first translate "function" into function, and gave the definition of "every formula contains heaven, which is a function of heaven". So the function we are studying today is thanks to these mathematicians who have contributed to mathematics.

We learned the concept of function in junior high school: in the process of change, there are two variables X and Y. If an X value is given and a Y value is determined accordingly, then we call Y a function of X, where X is the independent variable and Y is the dependent variable. The range of x is called the definition range, and the range of y is called the range.

Example (1) After a shell was fired, it landed and hit the target 26s later. The firing height of the projectile is 845m, and the variation law of the height h (unit: m) of the projectile from the ground with time t (unit: s) is h= 130t-5t? A={t|0≤t≤26}，B={h|0≤h≤845}

We find that for any time t in number set A, according to the corresponding relation h= 130t-5t? In number set B, there is a unique height h corresponding to it, which conforms to the definition of function and should be a function. It is found that analytic expressions can be used to describe functions.

Difference: Example (1) uses analytic expressions to describe the corresponding relationship between variables.

Correspondence between Example (2) and Image Characterization Variables

The correspondence between Example (2) and the variables described in the table.

* * * Similarity: ① There are two groups of non-empty numbers.

② There is a definite correspondence between two groups of numbers, that is, according to this correspondence, there is a unique definite number in set B for any number in set A. ..

Therefore, in order to explore the essence of function, we give a new definition of function from the perspective of set and correspondence.

1. Generally speaking, let a and b be non-empty number sets. If any number X in set A has a unique number f(x) corresponding to it according to a certain correspondence F, then f:A→B is called a function from set A to set B ... Note: y = f (x), x ∈ A.

Guide students to deeply understand the main points of the definition and the conditions they meet.

Key points: ① A function is firstly the correspondence between two data sets.

② For each value of X, there is a unique Y value corresponding to it according to a certain correspondence F, which should be one-to-one correspondence or one-to-many correspondence between numbers.

③ Understand the meaning of y = f (x) carefully: y = f(x) is a whole, and f(x) does not represent the product of f and x, but a symbol, which can be an analytical formula, such as example (1); It can also be an image, as in example (2); It can also be a table, as in Example (3); Y = f (x) Like a processing factory, the input number X is processed into another value Y according to some processing process such as analytical formula, image and table.

④x is called the independent variable, and the value range A of X is called the domain of the function.

Y is called the function value, and the range of value of y C={f(x)|x∈A} is called the range of value of function and c ≤ b.

Emphasize the definition domain, the value domain is a set and the value domain is a subset of set B.

These two definitions are essentially the same, that is, the meanings of their domain and value domain are exactly the same, and the essence of the corresponding relationship is the same, but the starting point of narration is different. The definition given by junior high school is from the perspective of movement change, in which the corresponding relationship is to correspond each value of independent variable X with the unique function Y; The definition given by senior high school is from the perspective of set correspondence, in which correspondence is to correspond any element in set A with the only determined element in set B, which makes the definition free from the bondage of physical movement and more perfect.

function

Definition of 1. function

2. The three elements of a function

3. Judge whether the two functions are equal.