"Representation of Functions" Concept and Properties of Functions PPT

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"Representation of Functions" Concept and Properties of Functions PPT

Part One: Explanation of Curriculum Standards

1. Master the three representation methods of functions: analytic method, list method, graphical method and their respective advantages and disadvantages. In the analytic method, especially master the use of substitution and substitution methods to find the analytic expression of the function.

2. In practical problems, be able to choose appropriate representations to represent functions.

3. Be able to use function graphs to find the value range of a function and determine the changing trend of the function value.

Representation of function PPT, part 2 content: independent learning

1. Representation of function

1. (1) How are the expression methods of the three commonly used functions learned in junior high school defined?

Tips: ① Analytical method: Use mathematical expressions to express the correspondence between two variables; ② Graphic method: Use images to express the correspondence between two variables; ③ List method: List tables to represent the two variables correspondence between.

(2) Questions 1 to 4 on pages 60 and 61 of the textbook, what methods are used to represent functions?

Tips: Questions 1 and 2 use the analytical method, question 3 uses the image method, and question 4 uses the list method.

(3) What are the advantages and disadvantages of each of the three representation methods of functions?

(4) Do it

A classmate plans to buy x(x∈{1,2,3,4,5}) 2B pencils. The price of each pencil is 0.5 yuan, and a total of y yuan is needed. Therefore, a functional relationship is established between y and x. .

①What is the domain of function?

Tip:{1,2,3,4,5}

②What is the relationship between y and x?

Tip:y=0.5x

③Use a table to express the relationship between y and x.

Tip: The table is as follows:

2. Image of function

1. (1) In junior high school, we have studied the images of straight lines, inverse proportional functions and quadratic functions. Please make images of y=2x-1, y= ,y=x2. Observe what common characteristics these images have?

Tip: The common feature is composed of points that meet certain conditions. Specifically, the independent variable x in the function y=f(x) is used as the abscissa and the corresponding dependent variable y is used as the ordinate. All points That is, the graph that constitutes the function.

(2) How to draw the graph of function y=f(x)?

Tip: Taking a value x0 of the independent variable as the abscissa, we get a point (x0, f(x0)) on the coordinate plane. When the independent variable takes every value of the function domain A, we get a series of such points. , the set (point set) composed of all these points is {(x,y)|y=f(x),x∈A}, and the curve composed of these points is the image of the function y=f(x) .

(3) How to judge whether an image represents a function of y with respect to x?

Tip: If you draw a straight line perpendicular to the x-axis, if this straight line has at least one intersection with the graph, the graph is the graph of a certain function.

(4) How to determine its domain and value range from the graph of a function?

Tip: The interval represented by the projection of the image on the x-axis is the definition domain, and the interval represented by the projection on the y-axis is the value range.

2. Do it

The following graphics can only represent the graph of the function y=f(x) ()

Answer:D

Representation of functions PPT, part 3: inquiry learning

list representation function

Example 1 (one question with multiple spaces) The known functions f(x) and g(x) are given in the following table:

Then f(g(1))=____; when g(f(x))=2, x=____.

Analysis: This is a function evaluation problem represented by the list method. When solving, just find the corresponding value of the variable.

Analysis: From the correspondence table of g(x), we know that g(1)=3, ∴f(g(1))=f(3).

From the correspondence table of f(x), we know that f(3)=1, ∴f(g(1))=f(3)=1.

From the correspondence table of g(x), we know that when x=2, g(2)=2.

And g(f(x))=2, ∴f(x)=2. And from the correspondence table of f(x), we know that when x=1, f(1)=2.∴x=1.

Answer: 1 1

Reflection: The list method is an important way to represent functions, just like the tables we list when drawing function images. Its obvious advantage is that the function values ​​corresponding to the variables can be found directly in the table without calculation.

Extended exploration Under the known conditions of this example, g(f(1))=________; when f(g(x))=2, x=________.

Analysis:∵f(1)=2,∴g(f(1))=g(2)=2.

∵f(g(x))=2,∴g(x)=1,∴x=3.

Answer: 2 3

Find the analytical expression of a function

Example 2 (1) It is known that f(x+1)=x2-3x+2, find f(x);

(2) It is known that f(x) is a quadratic function and satisfies f(0)=1, f(x+1)-f(x)=2x, find the analytical formula of f(x);

(3) It is known that function f(x) has f(x)+2f(-x)=3x-2 for any x, find f(x).

Analysis: (1) (Method 1) Let x+1=t, substitute x=t-1 into f(x+1)=x2-3x+2 to get f(t), you can get f(x); (Method 2) Since the position of x+1 in f(x+1) is the same as the position of x in f(x), f(x+1) can also be transformed into f(x+1)=(x+ 1)2-5(x+1)+6.(2) Suppose f(x)=ax2+bx+c(a≠0), and then list the system of equations according to the conditions to find the values ​​of a, b, c .(3) Replace x in f(x)+2f(-x)=3x-2 with -x and solve the system of equations about f(x) and f(-x).

Solution: (1) (Method 1) Let x+1=t, then x=t-1.

Substitute x=t-1 into f(x+1)=x2-3x+2,

We get f(t)=(t-1)2-3(t-1)+2=t2-5t+6,

∴f(x)=x2-5x+6.

(Method 2) ∵f(x+1)=x2-3x+2=x2+2x+1-5x-5+6=(x+1)2-5(x+1)+6,

∴f(x)=x2-5x+6.

(2) Suppose the quadratic function required is f(x)=ax2+bx+c(a≠0).

∵f(0)=1, ∴c=1, then f(x)=ax2+bx+1.

∵f(x+1)-f(x)=2x is true for any x∈R,

∴a(x+1)2+b(x+1)+1-(ax2+bx+1)=2x,

Function representation PPT, the fourth part: thinking analysis

Error caused by ignoring the actual meaning of variables

For example, in the rectangle ABCD, BA=3, CB=4, point P moves on AD, CQ⊥BP, Q is the vertical foot. Assume BP=x, CQ=y, try to find the functional expression of y with respect to x formula, and draw the graph of the function.

The wrong solution is based on the meaning of the question, and we get △CQB∽△BAP,

So CQ/BA=CB/BP,

That is y/3=4/x.

So y=12/x.

Therefore, the required function expression is y=12/x, and its graph is as shown in the figure.

What are the errors in the above problem-solving process? What are the reasons for the errors? How do you correct them? How to prevent them?

Tip: The actual significance of x was not considered in the above problem-solving process, thus expanding the value range of x and causing errors.

Representation of functions PPT, part 5: practice in class

1. It is known that the graph of a linear function passes through the points (1,0) and (0,1), then the analytical formula of the linear function is ()

A.f(x)=-x B.f(x)=x-1

C.f(x)=x+1 D.f(x)=-x+1

Analysis: Suppose f(x)=ax+b(a≠0), then we have {■(a+b=0"," @b=1"," )┤

So a=-1, b=1, that is, f(x)=-x+1.

Answer:D

2. One morning, Xiao Ming was riding to school. When he started, he felt that time was tight, and then accelerated forward. Later, he found that there was still plenty of time, so he slowed down. The image that best matches the above incident is ()

Analysis: Because the first paragraphs of options A and D both advance at a constant speed, which does not meet the meaning of the question, options A and D are eliminated. They first accelerate and then slow down, indicating that the speed of the image rises first and then slows down, so choose C.

Answer:C

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Update Time: 2024-09-06

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