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函数的奇偶性_图文

观察下图,思考并讨论以下问题:
(1) 这两个函数图象有什么共同特征吗? (2) 相应的两个函数值对应表是如何体现这些特征的?

f(x)=x2
f(-3)=9=f(3) f(-2)=4=f(2) f(-1)=1=f(1)

f(x)=|x|
f(-3)=3=f(3) f(-2)=2=f(2) f(-1)=1=f(1)

实际上,对于R内任意的一个x,都有f(-x)=(-x)2=x2=f(x), 这时我们称函数y=x2为偶函数.

1.偶函数
一般地,对于函数f(x)的定义域内的任意一个x, 都有f(-x)=f(x),那么f(x)就叫做偶函数.
2 例如,函数 f ( x) ? x ? 1, f ( x) ? x 2 ? 1 都是偶函数,
2

它们的图象分别如下图(1)、(2)所示.

观察函数f(x)=x和f(x)=1/x的图象(下图),你能发
现两个函数图象有什么共同特征吗?

f(-3)=-3=-f(3) f(-2)=-2=-f(2) f(-1)=-1=-f(1)

f(-3)=-1/3=-f(3) f(-2)=-1/2=-f(2) f(-1)=-1=-f(1)

实际上,对于R内任意的一个x,都有f(-x)=-x=-f(x),这时 我们称函数y=x为奇函数.

2.奇函数
一般地,对于函数f(x)的定义域内的任意一个x, 都有f(-x)= - f(x),那么f(x)就叫做奇函数. 注意: 1、函数是奇函数或是偶函数称为函数的奇偶性, 函数的奇偶性是函数的整体性质; 2、由函数的奇偶性定义可知,函数具有奇偶性的 一个必要条件是,对于定义域内的任意一个x,则 -x也一定是定义域内的一个自变量(即定义域关 于原点对称).

例5、判断下列函数的奇偶性:
(1) f ( x) ? x 4 1 (3) f ( x ) ? x ? x

( 2) f ( x) ? x 5 1 ( 4) f ( x ) ? 2 x

(1)解:定义域为R ∵ f(-x)=(-x)4=f(x) 即f(-x)=f(x)

(2)解:定义域为R f(-x)=(-x)5=- x5 =-f(x) 即f(-x)=-f(x)

∴f(x)偶函数 ∴f(x)奇函数 (3)解:定义域为{x|x≠0} (4)解:定义域为{x|x≠0} ∵ f(-x)=-x+1/(-x)=-f(x) ∵ f(-x)=1/(-x)2=f(x) 即f(-x)=-f(x) 即f(-x)=f(x)

∴f(x)奇函数

∴f(x)偶函数

课堂练习
判断下列函数的奇偶性:

1 (1) f ( x) ? x ? x (3) f ( x) ? 5 (5) f ( x) ? x ? 1

(2) f ( x) ? ? x ? 1
2

(4) f ( x) ? 0 (6) f ( x) ? x , x ? [?1,3]
2

已知f(x)是定义在R上的奇函数,当x≥0时, f(x)=x(1+x).求当x<0时,函数的解析式

解:当x ? 0时, -x ? 0 ? f (? x) ? ? x(1 ? x) ? x ? x
2

? f ( x)是R上的奇函数 ? f (? x) ? ? f ( x) ? ? f ( x) ? x ? x
2

即f ( x) ? ? x ? x, ( x ? 0)
2

用定义判断函数奇偶性的步骤:
(1)、先求定义域,看是否关于原点对称; (2)、再判断f(-x)=-f(x)或f(-x)=f(x)是否恒成立.

本课小结
1、两个定义:对于f(x)定义域内的任意一个x,

f(x)为奇函数 ? 如果都有f(-x)=f(x) ? f(x)为偶函数 如果都有f(-x)=-f(x)
2、两个性质:

一个函数为奇函数
一个函数为偶函数

它的图象关于原点对称 ? 它的图象关于y轴对称 ?

作业:


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