函數(shù)的奇偶性(含習(xí)題練習(xí))

1、函數(shù)的奇偶性1.函數(shù)奇偶性的定義及圖象特點奇偶性定義圖象特點偶函數(shù)如果對于函數(shù)f x的定義域內(nèi)任意一個x ,都有f x f x ,那么函數(shù)f x是偶函數(shù)圖象關(guān)于y軸對稱奇函數(shù)如果對于函數(shù)f x的定義域內(nèi)任意一個x ,都有f xf x ,那么函數(shù)f x是奇函數(shù)圖象關(guān)于原點對稱判斷f ( x)與 f x的關(guān)系時， 也可以使用如下結(jié)論： 如果f ( x) f x 0或f( x)1( f (x) 0),則函數(shù)f x為偶函數(shù)；如果f ( x) f x 0或 f (x)f ( x)1( f (x) 0),則函數(shù)f x為奇函數(shù).f (x)注意：由函數(shù)奇偶性的定義可知，函數(shù)具有奇偶性的一個前提條件是：對于定義

2、域內(nèi)的任意一個x,x也在定義域內(nèi)(即定義域關(guān)于原點對稱).定義：設(shè) y f (x) , x A,如果對于任意 x A,都有f ( x) f (x),則稱y f (x)為偶函數(shù)。如果對于任意x A,都有f ( x)f (x),則稱y f (x)為奇函數(shù)。證明：任意一個定義域關(guān)于原點對稱的函數(shù)均可以寫為一個奇函數(shù)和偶函數(shù)之和且唯一。若函數(shù) f x 的定義域關(guān)于原點對稱，則 f x 可以表示為f x 1 f x f x 1 f x f x ,該式的特點是：右端為一個奇函數(shù)和一個偶 22函數(shù)的和。2.性質(zhì)：y f (x)是偶函數(shù)y f (x)的圖象關(guān)于y軸對稱；y=f(x)是奇函數(shù)y f (x)的圖象

3、關(guān)于原點對稱。若奇函數(shù)定義域中有0,則必有f (0)0.即0 f (x)的定義域時，f (0)0是f (x)為奇函數(shù)的必要非充分條件.對于偶函數(shù)而言有：f ( x) f (x) f (| x|)。既奇又偶函數(shù)有無窮多個(f (x) 0 ,定義域是關(guān)于原點對稱的任意一個數(shù)集)。奇±奇二奇偶±偶才禺奇x奇才禺偶*偶=偶奇x偶的兩函數(shù)白定義域D, D2, DAD也關(guān)于原點對稱 奇函數(shù)的導(dǎo)函數(shù)為偶函數(shù)，偶函數(shù)的導(dǎo)函數(shù)為奇函數(shù)。復(fù)合函數(shù)奇偶性質(zhì)y f (x 1)為偶函數(shù)f (x 1) f ( x 1) f (x)關(guān)于 x 1 軸對稱y f (x 1)為奇函數(shù)f (x 1)f ( x

4、1) f (x)關(guān)于(1,0)點對稱y f (x)為偶函數(shù)f (x 1) f ( x 1) f (x)關(guān)于y軸軸對稱y f (x)為奇函數(shù) f (x 1)f ( x 1) f (x)關(guān)于原點點對稱。y f (x)為奇函數(shù)f (x 1)關(guān)于(1, 0)點對稱y f (x)為偶函數(shù)f (x 1)關(guān)于x 1軸對稱奇函數(shù)在關(guān)于原點對稱的區(qū)間上的單調(diào)性相同，偶函數(shù)在關(guān)于原點對稱的區(qū)間上的單調(diào)性相反.f (x) , g(x)在它們的公共定義域上有下面的結(jié)論:f (x)g(x)f (x) g(x)f (x) g(x)f (x)g(x)f (g(x)偶函數(shù)偶函數(shù)偶函數(shù)偶函數(shù)偶函數(shù)偶函數(shù)偶函數(shù)奇函數(shù)不能確定不能

5、確定奇函數(shù)偶函數(shù)奇函數(shù)偶函數(shù)不能確定不能確定奇函數(shù)偶函數(shù)奇函數(shù)奇函數(shù)奇函數(shù)奇函數(shù)偶函數(shù)奇函數(shù)若函數(shù)y f x的定義域關(guān)于原點對稱，則 f x f x為偶函數(shù)，f x f x為奇函數(shù)，f x f x為偶函數(shù).4.常見的奇函數(shù)及偶函數(shù)x2n1為奇函數(shù)，x2n為偶函數(shù)，多項式函數(shù) f （x）,僅含有奇數(shù)次項時為奇函數(shù)，僅含有偶數(shù)次項及常數(shù)項時為偶函數(shù)，既含有奇數(shù)次項也含有偶數(shù)次項時非奇非偶。 sin kx ,ax 1logalog |1a函數(shù)函數(shù)函數(shù)x為偶函數(shù)，ftan kx為奇函數(shù),1 cx1 cxx f x為奇函數(shù),cos kx為偶函數(shù)。為偶函數(shù)。2 xaxa1 ，一一1為奇函數(shù),1為奇函數(shù),2

6、2 xax a x、亙一為偶函數(shù)。axax 1ax 1ax 111-,-為奇函數(shù)。ax 1 2 ax 1 2log1 cx為奇函數(shù), a 1 cx二 log (12 aaxa xx x1 x loga .1 xloga x2 xa2 xa10gaJx為偶函數(shù)。(a 0 且 a2 2 cx）為奇函數(shù)。 c x '1）為奇函數(shù).1）為奇函數(shù).0且a 1）為奇函數(shù).xa1為奇函數(shù),ax 15 / 4你| x a |為奇函數(shù)，| x a | | x a |為偶函數(shù)。【典型例題】a x1 . (1)已知函數(shù)f xlog2 i-x為前函數(shù) 則頭數(shù)a的值為(2)函數(shù) f x lnx Jx21 ,且

7、f a f b 20,求 a b的值.2 .已知f (x) , g(x)分別是定義在R上的偶函數(shù)和奇函數(shù)，且f (x) g(x) x3 x2 1,則 f (1) g(1)3 .已知函數(shù)f (x)對任意的x R滿足f x f x ,且當(dāng)x 0時，f xx2 ax 1.若f x有4個零點，則實數(shù)a的取值范圍是 .4 .已知f(x)是R上的奇函數(shù)，且x 0時，f x 1,則不等式f x2 x f 0的解集 為.5 .已知f (x)為偶函數(shù)，當(dāng)x 0時，f (x) ex 1 x,則曲線y f (x)在點(1,2)處的切 線方程是6.若關(guān)于x的函數(shù)f xtx2 2x t 2 sinxx2 t0)的最大值為M,最小值為N,且M N4,則實數(shù)t的值為7.設(shè)函數(shù)f (x) ln(1 | x |)，則使得f (x)1 xf (2x 1)成立的x的取值范圍是()A.1 ,1