"Basic Properties of Functions" Concept and Properties of Functions PPT (Concept of Parity of Functions in Lesson 3)

"Basic Properties of Functions" Concept and Properties of Functions PPT (Concept of Parity of Functions in Lesson 3)

Part One: Learning Objectives

Combined with specific functions, understand the meaning of function parity and master the method of judging function parity.

Understand the relationship between function parity and symmetry of the graph of a function

Can use the parity of functions to solve simple problems

Basic properties of functions PPT, part 2: independent learning

Problem guide

Preview textbooks P82-P84 and think about the following questions:

1. What are the definitions of odd and even functions?

2. What are the characteristics of the domain of odd and even functions?

3. What are the characteristics of the graphs of odd and even functions?

A preliminary exploration of new knowledge

1. even function

(1) Definition: Generally speaking, assuming the domain of function f(x) is I, if ∀x∈I, both have _________, and __________________, then function f(x) is called an even function.

(2) Image features: The image is symmetrical about _________.

2. odd function

(1) Definition: Generally speaking, assuming the domain of function f(x) is I, if ∀x∈I, both have _________, and ______________, then function f(x) is called an odd function.

(2) Image features: The image is symmetrical about _________.

■Instructions from famous teachers

(1) Characteristics of the definition domains of odd and even functions

Since f(x) and f(-x) must be meaningful at the same time, the definition domains of odd and even functions are symmetrical about the origin.

(2) Characteristics of the correspondence between odd and even functions

①The odd functions are f(-x)=-f(x)⇔f(-x)+f(x)=0⇔f(-x)f(x)=-1(f(x)≠0);

②The even function has f(-x)=f(x)⇔f(-x)-f(x)=0⇔f(-x)f(x)=1(f(x)≠0).

(3) Three concerns about function parity

①If an odd function is defined at the origin, then f(0)=0. Sometimes this conclusion can be used to deny that a function is an odd function;

② There is only one type of function that is both an odd function and an even function, that is, f(x)=0, x∈I, where the domain I is a non-empty set that is symmetric about the origin;

③ Functions can be divided into odd functions, even functions, both odd and even functions, and non-odd and non-even functions according to their parity.

self-test

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

(1) The domains of odd and even functions are symmetric about the origin. ()

(2) The graph of function f(x)=x2 is symmetrical about the origin. ()

(3) For the function f(x) defined on R, if f(-1)=-f(1), then the function f(x) must be an odd function. ()

(4) If f(x) is an odd function defined on R, then f(-x)+f(x)=0.()

Which of the following functions is an odd function ()

A． y＝|x| B. y＝3－x

C. y=1x3 D． y=-x2+14

If the function y=f(x), x∈[-2, and a] is an even function, then the value of a is ()

A． -2 B. 2

C. 0D. Can not be sure

Basic properties of functions PPT, the third part: interactive teaching and practice

Judgment of function parity

Determine the parity of the following functions:

(1)f(x)＝|x＋1|－|x－1|;

(2)f(x)＝x2－1＋ 1－x2;

(3)f(x)=1-x2x;

(4)f(x)=x+1, x>0, -x+1, x<0.

regular method

Two methods to determine the parity of a function

(1)Definition method

(2)Image method

[Note]The judgment of the parity of a piecewise function should be discussed in pieces, and attention should be paid to obtaining the corresponding analytical formula of the function according to the range of x.

Graphs of odd and even functions

It is known that the function y=f(x) is an even function defined on R, and when x≤0, f(x)=x2+2x. The graph of the function f(x) on the left side of the y-axis is now drawn, as the picture shows.

(1) Please complete the graph of the complete function y=f(x);

(2) Write the increasing interval of function y=f(x) based on the image;

(3) Based on the image, write the set of x values ​​that make f(x)<0.

regular method

Steps to skillfully use parity and evenness to plot function graphs

(1) Determine the parity of the function.

(2) Draw the corresponding image of the function on [0, +∞) (or (-∞, 0]).

(3) According to the symmetry of the odd (even) function about the origin (y-axis), the corresponding function graph on (-∞, 0° (or 0, +∞)) can be obtained.

[Note] When making a symmetrical image, you can start from the symmetry of the point. The symmetrical point of the point (x0, y0) about the origin is (-x0, -y0), and the symmetrical point about the y-axis is (-x0, y0)．

Basic properties of functions PPT, Part 4: Feedback on achievement of standards

1. Which of the following functions is an even function ()

A． y=x

B． y=2x2-3

C. y=x

D. y＝x2，x∈(-1，1]

2. The graph of function f(x)=1x-x is about ()

A． y-axis symmetry

B． The straight line y=-x is symmetrical

C. Coordinate origin symmetry

D. The straight line y=x is symmetrical

3. It is known that the function f(x) is an odd function on R, and when x>0, f(x)=x2+1x, then f(-1)=________.

4. Based on the parity of the function in the question and the given partial image, draw the image of the function on the other side of the y-axis and solve the problem:

(1) Figure ① is a partial image of the odd function y=f(x), find f(-4)·f(-2);

(2) Figure ② is a partial image of the even function y=f(x). Compare the sizes of f(1) and f(3).

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