"Function and its Representation Method" PPT courseware on the concept and properties of function (Lesson 2 Representation Method of Function)

Description

"Function and its Representation Method" PPT courseware on the concept and properties of function (Lesson 2 Representation Method of Function)

Part One: Learning Objectives

1. Master the three representation methods of functions: analytical method, image method, and list method. (emphasis)

2. Will choose the appropriate method to represent functions according to different needs. (difficulty)

3. Understand the concept of piecewise function, be able to find the function value of piecewise function, and draw the image of piecewise function. (main difficulty)

4. Be able to choose appropriate methods to express the functional relationship between two variables in practical problems, and be able to solve related problems. (main difficulty)

core competencies

1. Cultivate intuitive imagination literacy through the image method of functional representation.

2. Cultivate operational literacy through finding analytical expressions of functions.

3. Use functions to solve practical problems and cultivate mathematical modeling literacy.

Functions and their representation PPT, part 2: independent preview and exploration of new knowledge

A preliminary exploration of new knowledge

1. image of function

(1) Definition: The independent variable x and the corresponding function value y in the function y=f(x), x∈A are respectively regarded as the abscissa and ordinate of the point in the plane rectangular coordinate system, then the point that meets the conditions The set F composed of (x, y) is called the image of the function, that is, F={(x, y)|y=f(x), x∈A}.

(2)F is the image of the function y=f(x), which must satisfy the following two conditions

①The coordinates (x, y) of ______ on the image all satisfy the functional relationship y=f(x);

②The points (x, y) that satisfy the functional relationship y=f(x) are all in ______.

2. function representation

Thinking 1: Can any function be expressed in three forms: analytical method, list method, and image method?

Tip: Not necessarily.

Not all functions can be expressed by analytical expressions. Not only that, the image method is not applicable to all functions, such as D(x)=0, x∈Q, 1, x∈∁RQ. Although the list method is theoretically applicable For all functions, but for situations where the independent variable has countless values, the list method can only represent an overview or fragment of the function.

3. piecewise function

If a function, within its domain, has different ________ for different _________ of independent variables, it is called a piecewise function.

Thinking 2: Is the piecewise function one function or several functions?

Tip: A piecewise function is a function, not several functions.

First try

1. It is known that the function f(x) is given by the following table, then f(3) is equal to ()

x 1≤x<2 2 2

f(x) 1 2 3

A.1　B． 2

C. 3D． does not exist

2. The vertex of the image of the quadratic function is (0, -1), and the symmetry axis is the y-axis. Then the analytical formula of the quadratic function can be ()

A． y=-14x2+1 B. y=14x2-1

C. y=4x2-16 D. y=-4x2+16

3. The formula given below is a piecewise function ()

①f(x)＝x2+1, 1≤x≤5, 2x, x<1.

②f(x)＝x+1, x∈R, x2, x≥2.

③f(x)＝2x+3, 1≤x≤5, x2, x≤1.

④f(x)=x2+3, x<0, x-1, x≥5.

A． ①② B. ①④

C. ②④ D． ③④

Functions and their representation PPT, the third part: cooperative exploration to improve literacy

Three ways to represent functions

[Example 1] 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 and the number of payments y. Use the list method, image method, and analytical method to express them ．

[Solution] ①The list method is as follows:

②Image method: As shown in the figure.

③Analytical method: y=3 000x, x∈{1,2,3,…,10}.

regular method

The list method, image method and analytical method describe the corresponding relationship between independent variables and function values ​​from three different perspectives. The same function can be expressed in different ways. When using the three methods to express the function, please pay attention to: ① The analytical method must Indicate the domain of the function; ② The independent variables selected in the list method must be representative and should reflect the characteristics of the domain; ③ In the image method, pay attention to whether there are lines.

How to find the analytical expression of a function

[Example 2] (1) It is known that f(x+1)=x-2x, find the analytical formula of f(x);

(2) It is known that the function f(x) is a linear function. If f(f(x))=4x+8, find the analytical formula of f(x);

(3) As shown in the figure, it is known that the isosceles trapezoid ABCD with a base angle of 45°, the length of the base BC is 7 cm, and the length of the waist is 22 cm. When the straight line l is perpendicular to the base BC (the vertical foot is F) When moving from left to right (having a common point with the trapezoid ABCD), the straight line l divides the trapezoid into two parts. Let BF=x, try to write the functional analytical formula of the area y of the left part with respect to x.

[Ideas Enlightenment] (1) Use substitution method or matching method to solve; (2) Use undetermined coefficient method to solve; (3) According to the position of point E, E can be divided into segment AB and E on segment Segments AD and E are solved separately for three types of line segments CD.

regular method

Common methods for finding analytical expressions of functions

1 Undetermined coefficient method: If the type of the analytical expression of f(x) is known, set its general form and determine the relevant coefficients based on the special values.

2 Substitution method: Suppose t=g(x), solve for x, substitute f(g(x)), and find the analytical formula of f(t).

3 Matching method: Match and transform the analytical expression of f(g(x)) so that it can be expressed by g(x), and then use x to replace all "g(x)" on both sides.

4. System of equations method  or elimination method : When two elements in the same correspondence are inverse or reciprocal to each other, a system of equations can be constructed to solve the problem.

Reminder: 1. When applying the substitution method to find the analytical expression of a function, be sure to ensure the equivalence of the function before and after substitution.

2 In the context of practical problems, the independent variable value ranges are different and the corresponding relationships are also different. In this case, it needs to be represented by a piecewise function.

Class summary

1. There are three commonly used expression methods for functions. You can choose at the right time to express the function in the best way. If you do not indicate the domain after the analytical expression, it can be regarded as the domain of the function as the set of real numbers R or that makes the analytical expression meaningful. A subset of R.

2. When constructing a function image, the image must reflect the extension direction of the image, whether there is any intersection with the x-axis and y-axis, whether the image is symmetrical, and special points must be marked.

3. A piecewise function is one function, not several functions.

4. To evaluate a piecewise function, you must first find out the interval in which the independent variable is located; the domain and value range of the piecewise function are the union of the domain and value range of each piecewise function respectively.

Functions and their representation methods PPT, the fourth part: reaching the standard in class and solidifying the double base

1. Thinking and analysis

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

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

(3) The piecewise function consists of several functions. ()

(4) The function f(x)=x+1, x≤1, -x+3, x>1 is a piecewise function. ()

2. Assume function f(x)=x2+1, x≤1, 2x, x>1, then f(f(3))=()

A.15B. 3

C.23D.139

3. The graph of function y=f(x) is as shown in the figure, and its analytical formula is ________.

4. It is known that the function f(x)=x2-2x(-1≤x≤2).

(1) Draw a simplified diagram of the f(x) image;

(2) Write the value range of f(x) based on the image.

Keywords: Free download of PPT courseware for compulsory course No. 1 Mathematics of High School People's Education B version, function and its representation method PPT download, concept and properties of function PPT download, representation method of function PPT download, .PPT format;

For more information about the PPT courseware "The Concept and Properties of Functions, Functions and Their Representation Methods, Representation Methods of Functions", please click the Concept and Properties of Functions ppt Functions and Their Representation Methods ppt Representation Methods of Functions ppt tag.

"Parity of Functions" Concept and Properties of Functions PPT (Application of Parity in Lesson 2):

"Parity of Functions" Concept and Properties of Functions PPT (Application of Parity in Lesson 2) Part One: Learning Objectives 1. Be able to find function values ​​or analytical expressions based on parity of functions. 2. Able to use the parity and monotonicity of functions to analyze and solve simpler problems..

"Parity of Functions" Concept and Properties of Functions PPT (Application of Parity in Lesson 2):

"Parity of Functions" Concept and Properties of Functions PPT (Application of Parity in Lesson 2) Part One: Learning Objectives 1. Be able to find function values ​​or analytical expressions based on parity of functions. 2. Able to use the parity and monotonicity of functions to analyze and solve simpler problems..

"Parity of Functions" Concept and Properties of Functions PPT (Lesson 1: The Concept of Parity):

"The Parity of Functions" PPT on the concepts and properties of functions (the concept of parity in Lesson 1) Part One Content: Learning Objectives 1. Understand the definitions of odd functions and even functions. 2. Understand the characteristics of the graphs of odd and even functions. 3. Master the method of judging the parity of functions..

File Info

Update Time: 2024-11-25

This template belongs to Mathematics courseware People's Education High School Mathematics Edition B Compulsory Course 1 industry PPT template

Preview