"The Concept and Representation of Functions" The Concept and Properties of Functions PPT (The Representation of Functions in the Second Lesson)

"The Concept and Representation of Functions" The Concept and Properties of Functions PPT (The Representation of Functions in the Second Lesson)

Part One: Learning Objectives

Understand the three representation methods of functions and their respective advantages and disadvantages, and be able to choose appropriate methods to represent functions according to different needs.

Master common methods for finding analytical expressions of functions

Ability to plot graphs of functions and obtain useful information from graphs

The concept of function and its representation PPT, part 2: independent learning

Problem guide

Preview textbook P67 and think about the following questions:

1. What are the representation methods of functions?

2. What are the characteristics of the representation method of functions?

A preliminary exploration of new knowledge

function representation

■Instructions from famous teachers

(1) List method: The premise of using the list method is that the function values ​​correspond clearly, and the selected independent variables must be representative.

(2) Image method: The image can be either a continuous curve or a discrete point.

(3) Analytical method: The premise of using the analytical method to express a function is that the correspondence between variables is clear, and when using the analytical method to express a function, attention should be paid to indicating its domain of definition.

self-test

Judge whether it is true or false (mark “√” if it is correct and “×” if it is wrong)

(1) Any function can be expressed analytically. ()

(2) The graph of the function must be a continuous curve on the defined interval. ()

It is known that y is inversely proportional to x, and when x=2, y=1, then the functional relationship of y with respect to x is ()

A． y＝1x　B． y=-x

C. y=2x D. y=x2

It is known that the function f(x) is given by the following table, then f(f(3))=________.

x 1 2 3 4

f(x) 3 2 4 1

The graph of function f(x) is as shown in the figure, then the domain of f(x) is ________ and the value range is ________.

The concept of function and its representation PPT, the third part: lecture and practice interaction

Three ways to represent functions

A shopping mall has newly purchased 10 color TVs, each priced at 3,000 yuan. Try to find the functional relationship between the number of units sold x (x is a positive integer) and the number of payments y. Use the list method, image method, and analysis respectively. The law is expressed.

regular method

(1) Choice of three representation methods for functions

The analytical method, the graphical method and the list method depict the corresponding relationship between independent variables and function values ​​from three different perspectives. The premise of using the analytical method is that the corresponding relationship between variables is clear, the premise of using the graphical method is that the changing rules of the function are clear, and the premise of using the list method is that the number of independent variables in the definition domain is small.

(2) When applying the three representation methods of functions, the following three points should be noted:

①The analytical method must indicate the domain of the function;

②The list method must be able to clearly show the corresponding relationship between independent variables and function values;

③In the image method, it must be clear whether the function image is a "point" or a "line".

Track training

1. A student left home to go to school. He started running and then walked the rest of the distance when he was tired. In the following figure, the vertical axis represents the distance from school, and the horizontal axis represents the time since departure. The one that is more consistent with the student's walking method is ()

Analysis: Choose D. From the meaning of the question, it can be seen that the speed is faster at the beginning, and then the speed becomes slower, so the initial curve is steeper, and then the curve is relatively gentle, and the vertical axis represents the distance from school, so the distance is the largest at the beginning, and the final distance is 0.

2. The following table shows that the function y=f(x), then the set of integer solutions of f(x)>x is ________.

x 0

y＝f(x) 4 6 8 10

Analysis: When 0x is {1, 2, 3}.

When 5≤x<10, the integer solution of f(x)>x is {5}.

When 10≤x<15, the integer solution of f(x)>x is ∅.

When 15≤x<20, the integer solution of f(x)>x is ∅.

To sum up, the set of integer solutions for f(x)>x is {1, 2, 3, 5}.

Find the analytical expression of a function

(1) It is known that f(x) is a linear function, and f(f(x))=9x+4, find the analytical formula of f(x);

(2) It is known that f(x+1)=x+2x, find f(x);

(3) It is known that 2f1x+f(x)=x(x≠0), find f(x).

[Solution] (1) Suppose f(x)=kx+b(k≠0),

Then f(f(x))=k(kx+b)+b=k2x+kb+b=9x+4.

So k2=9, kb+b=4.

The solution is k=3, b=1, or k=-3, b=-2.

So f(x)=3x+1 or f(x)=-3x-2.

regular method

Common methods for finding analytical expressions of functions

(1) Undetermined coefficient method: If the type of the function is known, the undetermined coefficient method can be used to solve the problem, that is, the analytical formula of the function is set up based on the function type, and then the equation (set) is listed according to the conditions, and the undetermined coefficient is obtained by solving the equation (set). Then find the analytical formula of the function.

(2) Substitution method (sometimes the "matching method" can be used): If the analytical formula of the function f(g(x)) is known, the analytical formula of f(x) can be found by the substitution method (or the "matching method"). That is, let g(x)=t, solve for x in reverse, and then substitute it into f(g(x)) to find f(t), thus f(x).

(3) Elimination method (or method of solving a system of equations): In the known formula, there are functions about two different variables, and these two variables have a certain relationship. In this case, the relationship between the two variables must be used. , establish a new formula about these two variables, establish a system of equations from the two formulas, eliminate one variable by solving the system of equations, and obtain the analytical formula of the target variable. This method is called the elimination method (or solving the system of equations Law).

The concept of function and its representation PPT, Part 4: Feedback on achievement of standards

1. The graph of the known function f(x) is as shown in the figure. The coordinates of points A and B are (0, 3), (3, 0) respectively, then f (f (0)) = ()

A． 2B. 4

C. 0D. 3

2. It is known that the function f(2x+1)=6x+5, then the analytical formula of f(x) is ()

A． f(x)=3x+2 B． f(x)＝3x＋1

C. f(x)=3x-1 D. f(x)＝3x＋4

3. It is known that the function f(x)=x-mx, and the graph of this function passes through the point (5, 4), then the value of the real number m is ________.

4. It is known that f(x) is a quadratic function and satisfies f(0)=1, f(x+1)-f(x)=2x, find f(x).

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