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"The Concept and Representation of Function" PPT courseware on the concept and properties of function (the third lesson piecewise function)
Part One: Learning Objectives
1. Understand the concept of piecewise function, be able to find the function value of piecewise function, and be able to draw the image of piecewise function. (main difficulty)
2. Be able to list piecewise functions in practical problems and solve related problems. (main difficulty)
3. Through the study of this section, students can understand the meaning of piecewise functions and improve their abilities in mathematical modeling and mathematical operations. (emphasis)
core competencies
1. Cultivate mathematical operation literacy through piecewise function evaluation problems.
2. Use piecewise functions to solve practical problems and cultivate mathematical modeling literacy.
The concept of function and its representation PPT, the second part: independent preview and exploration of new knowledge
A preliminary exploration of new knowledge
piecewise function
If a function y=f(x), x∈A has different corresponding relationships according to the different value ranges of the independent variable x in A, then such a function is called a piecewise function.
Thinking: Is the piecewise function one function or several functions?
Tip: A piecewise function is a function, not several functions.
First try
1. 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． ①②B. ①④
C. ②④ D． ③④
B[Combined with the definition of piecewise function, it can be seen that ①④ are piecewise functions, and the definition domains of different corresponding relationships in ②③ overlap, so B.]
2. The value range of the function y=x, x≥0, -x, x<0 is ________.
3. Function f(x)=x+1, x≤1, -x+3, x>1, then f(f(4))=________.
The concept of function and its representation PPT, the third part of the content: cooperative exploration to improve literacy
Piecewise function evaluation problem
[Example 1] It is known that the function f(x)=x+1, x≤-2, x2+2x, -2
(1) Find the values of f(-5), f(-3), ff-52;
(2) If f(a)=3, find the value of the real number a.
[ Solution f(-5)=-5+1=-4,
f(-3)＝(-3)2+2×(-3)＝3-23.
∵f－52＝－52＋1＝－32，
And -2<-32<2,
∴ff－52＝f－32＝－322＋2×－32＝94－3＝－34.
(2) When a≤-2, a+1=3,
That is, a=2>-2, which does not fit the meaning of the question, so discard it.
When -2
That is, a2+2a-3=0.
∴(a-1)(a+3)=0,
Solve to get a=1 or a=-3.
regular method
1. How to find the value of a piecewise function:
(1) Determine which interval the independent variable of the required value belongs to.
(2) Substitute the analytical expression in this section to evaluate until the value is obtained. When the form f(f(x0)) appears, it should be evaluated sequentially from the inside to the outside.
2. Steps to find the letter value when the function value is known:
(1) First, classify and discuss the value range of letters.
(2) Then substitute it into different analytical formulas.
(3) Find the value of the letter by solving the equation.
(4) Check whether the value sought is within the interval in question.
Reminder: When finding the value of an independent variable under certain conditions, first assume that the value you want is in each segment of the defined interval of the piecewise function, and then find the value of the independent variable accordingly. Remember to substitute the test.
Class summary
1. A piecewise function is one function, not several functions.
2. 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.
3. graph of piecewise function
A piecewise function has several segments, and its graph consists of several curves. In the same rectangular coordinate system, draw images in sequence according to the definition interval and expression of each segment of the piecewise function. Pay attention to determine whether the endpoint of each segment of the image is a hollow point or a solid point. The images of each segment of the function are combined together. The graph of the entire piecewise function can be obtained.
The concept of function and its representation PPT, the fourth part: reaching the standard in class and solidifying the double base
1. Thinking and analysis
(1) The piecewise function consists of several functions. ()
(2) 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.23 D.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 f(x)=x2, -1≤x≤1, 1, x>1 or x<-1.
(1) Draw the image of f(x);
(2) Find the domain and value range of f(x).
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