Teacher: What is the inverse function? Let's think about this question together: in a function, if Y is taken as the independent variable and the dependent variable, can it form a function? Students can form a function. Teacher: Why is it a function? Any value within the allowable range of health: y has a unique X corresponding to it according to the law →. According to this classmate, this is in line with the definition of function, that is to say, according to the above principle, function has inverse function. What is the analytical expression of this inverse function? Student: I think so. Teacher: There is no problem with this expression, but it doesn't conform to our habits. According to the custom, the letter X is used to represent the independent variable and the letter Y is used to represent the dependent variable. So the analytical expression of this function can be written as such a change, which brings the question, that is, is sum the same function? Student: Yes. Teacher: Can you explain it in detail? Health: From the perspective of the three elements of a function, they have the same domain and value, both of which are R. At the same time, the corresponding law is that the dependent variable is obtained by subtracting 1 dividing the independent variable by 2, which is the same, so it is the same function. Teacher: Since they are the same, we call it the inverse function of the function. Similarly, does the function y = x-12 have an inverse function? Student: Yes, that's right. Teacher: In other words, function and function are reciprocal functions. So, do all functions have reciprocal functions? Student: Not all functions have inverse functions. Teacher: Can you give an example? Student: As a function, take Y as the independent variable and X as the dependent variable. Within the allowable range of Y, one Y may correspond to two X's, such as y= 1 and X = 1, so it can't form a function, which means there is no inverse function. Teacher: That's good. If you explain it from a formal point of view, you will see it more clearly, as shown in figure 1. Understand what the inverse function is, and summarize the previous research process of the inverse function of function y = 2x+ 1, and we can get the definition of the inverse function. Because this definition is long, let's take a look at the relevant content in the book. (Write on the blackboard: (1) Definition of inverse function) Ask the students to open the second paragraph on page 60 of the book and ask a classmate to read it aloud. In order to help students understand the description in the definition, teachers can take a specific function as an example to explain the relationship between y=f(x) and x=j(y). At the same time, it should be pointed out that the meaning of the word "if" in the definition means that not all functions have inverse functions. ) After having a preliminary understanding of inverse function, let's further study this special function concept. (blackboard writing: (2) understanding of concepts. ) teacher: inverse function. Let's take the sum of two functions y=2x+ 1 as an example. Student: The corresponding rules are different. Teacher: Can you be more specific? What is the difference? Student: In the corresponding rules of these two functions, the positions of X and Y are reversed. The relationship between the two functions should be studied from the perspective of the three elements of the function, and the teacher can appropriately guide the students to get closer to the three elements. ) teacher: is there any connection? Student: when's domain and value domain are the domain and domain of y=2x+ 1 respectively. Teacher: According to our discussion just now, we can find that the three elements of the inverse function are determined by the original function. After the given function is determined, the three elements of its inverse function are also determined, which can be called "three determinations" for short. To concretize this definite relation is the inverse of the inverse function. Health: the domain of the inverse function is the domain of the original function; The range of the inverse function is the domain of the original function; The inverse function correspondence rule is to exchange the positions of X and Y in the original function correspondence rule. Teacher: It can be seen that the "inverse" of the inverse function is actually embodied as "three inverses". In this "three inverses", it is the inverses of X and Y that play a decisive role. It is precisely because of the change of their positions that the corresponding numerical values are reversed, resulting in the other two inversions. (On the blackboard: A. If "three" has an inverse function, what is its nature? What is the relationship with the nature of the original function? Through the previous examples, we can find that the nature of the original function plays a decisive role in the above problems, and the nature of the inverse function is also related to the nature of the original function. Because of the close relationship between function and inverse function, it has become an important aspect of further study of function. When we study the properties of a function, if it has an inverse function, we can choose between the two and study it, which increases the research methods of the function. Teacher: We have a more comprehensive understanding of the concept of inverse function. Let's look at these two questions together. Example 1 inverse function. Student: (blackboard writing) The solution is the cause, so the inverse function is (if the expression is not standardized, don't pursue it for the time being, and then comment together after Example 2 is solved. ) student: (blackboard writing) solution is the reason for getting y=. Teacher: Let the students state these two examples. Student: The domains of the two functions are x≥ 1 and x≥2 respectively, so they are two different functions. Teacher: Why (x≥2)? Student: Because the definition domain of the inverse function should be the value domain of the original given function f(x), and the value domain of f(x) should be y≥2, so the inverse function should be (x ≥ 2). Teacher: Good point. According to our understanding of the inverse function, the domain of the inverse function is the range of the original given function. Therefore, to require the domain of the inverse function, we must first find the original domain. Student: Yes, and x≥ 1, so. Because of the range of, (x ≥ 2). Teacher: Through the discussion just now, we found and solved the problem of inverse function in Example 2. At the same time, we also notice that the domain of the inverse function must be clearly pointed out to ensure the correctness of the conclusion. Besides, are there any other questions? Health: Why didn't you find the range of the original function in the example 1? Teacher: Please discuss this problem. Student: Because the range of the given function is y≠0, which is consistent with the conclusion that the range of the inverse function is x≠0, there is no mistake. Teacher: The consistency of this conclusion in this question should be said to be accidental, not inevitable. Therefore, in the process of finding the inverse function, students must be asked to: (write on the blackboard) understand the reason, so finding the inverse function as a teacher: Can you summarize the basic steps of finding the inverse function of a function expressed by an analytical formula through the discussion of two specific examples just now? (Written on the blackboard: 2. Student: First, solve x from the analytical formula, then find the range of the given function, and finally rewrite it into a conventional expression. Teacher: Summarize these steps in a few simple sentences: 1. Inverse solution: that is, the analytical expression is regarded as an equation of x, and the analytical expression of the inverse function is obtained; 2. Exchange: Find the range of the given function and replace it with the definition range of the inverse function; 3. Rewrite: Write a function in one form. (Blackboard: 1. Inverse solution 2. Exchange 3. Rewrite. ) Teacher: After a few exercises, let's see if the students really understand these three basic steps. 3. Consolidate the exercise to find the inverse function of the following function. 1. (A student finishes on the blackboard. ) The solution is x=3 2y-2. F (x) = 4), so f- 1 (x) = 32x-2, x ∈ (-∞, 4) .2.y = x2-x+1(x ≥12) (by a student) Therefore, the range of x = 1 4y-32 and y=x2-x+ 1(x≥ 12) is {y | y ≥ 34}, so f- 1 (x) 1. Comment on the students' statements on the blackboard and the questions in other students' answers. ) Teacher: Look at the students' statements on the blackboard first. Please correct them. (A student corrects on the blackboard) From y=x2-x+ 1, x2-x+ 1-y=0, so X = 1. Therefore, the range of x= 1+4y-32 and y=x2-x+ 1(x≥ 12) is {y | y ≥ 34}, so the inverse function is y =1+4x-32 (x Some precautions. (1) One step in the process of finding the inverse function is to find the range of the original given function. There are many ways to evaluate the scope. If the given function is a common function, such as linear function and quadratic function, it may be more convenient and intuitive to evaluate the range from the perspective of "shape" (2) There are two roots in solving the quadratic equation of one variable about X, and X must be selected according to the conditions given in the title. (3) There are differences in the use of anti-function symbols between the two topics. If the title gives the symbol of f(x), the inverse function can be expressed by f- 1(x), otherwise it can only be described in words. Fourth, the summary is 1. Inverse function is an important concept in function, which is based on the study of the relationship between two functions. Therefore, understanding it should be studied from the perspective of three elements. 2. Whether a function has an inverse function is determined by the nature of the original function, and the nature of the inverse function is also determined by the nature of the original function. 3. Finding the inverse function is actually doing two things, one is to solve an equation about the independent variable X, and the other is to find the range of a function. 5. Exercise P65, Exercise 6, Question 3 (1), in the homework textbook. Question 4. The classroom teaching design shows that inverse function is a concept course, so the key to the success of this course is the establishment of the concept of inverse function. Inverse function is a special phenomenon in function. The purpose of learning this concept is to deepen and improve the understanding of the concept and nature of function, so students have a certain knowledge base and cognitive basis for learning this knowledge, so students' subjective participation should be the main line. But also the thinking and participation led by teachers. Students' thinking begins with problems, so the starting point of this class should be a problem with a large space for thinking. Therefore, in the design, the principle of providing research inverse function from a specific function is chosen, so that students can choose their own research methods and explore for themselves. In the process of research, students should be given timely and appropriate guidance in view of the obstacles they encounter. Guide students' thinking to the right track. The key to the establishment of the concept of inverse function is to let students understand from the perspective of the relationship between two functions, so as to deepen their understanding of the concept of function. In teaching design, teachers use concrete examples to help students find the most familiar knowledge and the most obvious examples of research methods, and then gradually summarize and abstract the meaning of inverse function, which is also convenient to disperse difficulties. Highlight the key points. The understanding of a concept is often reflected by a specific operation, and the flexibility and proficiency of the operation can also reflect the depth of understanding of the concept. Therefore, the understanding of the concept of inverse function in this course ultimately lies in the formation and training of the skills of finding inverse function. In the design, teachers adopt the basic steps of letting students try, adjust, summarize and finally form the inverse function. In practice, students are encouraged to try boldly and are not afraid of failure. In the process of learning knowledge, lessons are sometimes more profound than experience. In the teaching design of this class, students should be able to think, ask questions, analyze and solve problems actively from beginning to end, and in the process of active thinking, students' mathematical ability and mathematical literacy should be continuously improved.