"The Concept and Representation of Functions" The Concept and Properties of Functions PPT (Lesson 3 Piecewise Functions)

Description

"The Concept and Representation of Functions" The Concept and Properties of Functions PPT (Lesson 3 Piecewise Functions)

Part One: Learning Objectives

Understand the concept of piecewise function and be able to find the function value of piecewise function

Able to draw graphs of piecewise functions and apply them to solve problems

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

Problem guide

Preview the textbook P68-P71 and think about the following questions:

1. What is a piecewise function?

2. Is a piecewise function one function or multiple functions?

A preliminary exploration of new knowledge

1. 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.

■Instructions from famous teachers

(1) The piecewise function is one function, not several functions. When dealing with piecewise function problems, it is necessary to first determine in which interval the value of the independent variable lies, and then select the corresponding correspondence.

(2) When writing a piecewise function, curly brackets should be used to combine each piece of function into one function, and the value range of the independent variable of each piece of function must be specified.

(3) The domain of a piecewise function is the union of the value intervals of all independent variables. The domain of a piecewise function can only be written in the form of a set, and cannot be written separately in the form of several sets.

(4) The value range of a piecewise function is the union of the value ranges of each piecewise function within the value range of the corresponding independent variable.

2. 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. Pay attention to whether the endpoint of each segment is a hollow point or a solid point. When combined together, the image of the entire piecewise function is obtained.

■Instructions from famous teachers

When drawing the graph of each segment of a function, you can first use dotted lines to draw its image regardless of the limitations of the definition domain, and then use solid lines to retain its corresponding image within the defined interval of the segment, that is, "segment drawing" ．

self-test

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

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

(2) The function f(x)=1, x≥0, -1, x<0 is a piecewise function. ()

(3) The domain of a piecewise function is the union of the values ​​of the independent variables on each segment. ()

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

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

Domain and value range of piecewise functions

(1) It is known that the function f(x)=|x|x, then its domain is ()

A． R.B. (0,+∞)

C. (-∞,0) D. (-∞,0)∪(0,+∞)

(2) The definition domain of function f(x)=-x2+1, 0

regular method

(1) How to find the domain and value range of piecewise functions

①The definition domain of a piecewise function is the union of the definition domains of each piecewise function;

②The value range of a piecewise function is the union of the value ranges of each piecewise function.

(2) The definition domain and value range of absolute value functions usually need to be converted into piecewise functions to solve.

Piecewise function evaluation problem

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

f(-3), the value of ff-52.

(Variable question method) The conditions of this example remain unchanged. If f(a)=3, find the value of the real number a.

regular method

(1) Method of calculating function value of piecewise function

① Determine which interval the independent variable of the required value belongs to;

② Substitute the analytical expression of this section for evaluation 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 if the function value is known

① First, classify and discuss the value range of letters;

②Then substitute it into different analytical formulas;

③Find the value of the letter by solving the equation;

④ Check whether the value sought is within the interval in question.

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

1. The value range of function f(x)=y=2x2, 0≤x≤1, 2, 1

A． RB. [0，+∞)

C. [0,3] D． {y|0≤y≤2 or y=3}

2. It is known that the function y=x2+1, x≤0, -2x, x>0, then the value of x that makes the function value 5 is　()

A． -2 B. 2 or -52

C. 2 or -2 D． 2 or -2 or -52

3. The graph of function y=x+|x|x is ()

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Update Time: 2024-11-22

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

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