編稿 xf(-x)=f(x)，f(x)稱為偶函數(shù).xf(-x)=-f(x)，f(x)稱為奇函數(shù). （3）f(-x)=f(x)f(xf(x)0,f(x)1f(x)0ff(-x)=-f(x)f(xf(x)0f(x)1f(x)0，ff(x的定義域，判斷函數(shù)的定義域是否關于原點對稱，若不關于原點對稱，則該函數(shù)既f(xf(xf(xf(xf(xf(x的奇偶性.若f(x)=-f(x)，則f(x)是奇函數(shù)；f(xf(xf(xf(x)f(x)f(xf(x)f(xf(xf(xf(x既是奇函數(shù)，又是偶函數(shù)若函數(shù)的定義域是關于原點對稱的，再判斷f(x)與f(x)之一是否相等.f(xf(xf(x)f(x=0f(x)1ff(xf(xx與xf(xf(x對應不同的表達式，而它們的結果按奇偶函數(shù)的定奇函數(shù)在其對稱區(qū)間[a,b]和[-b，-a]上具有相同的單調性，即已知f(x)是奇函數(shù)，它在區(qū)間[a,b]-a]f(x是偶函數(shù)且在區(qū)間[a,b]（減函數(shù)f(x在區(qū)間[-1.1-1f(x)1-1

(2)f(x)=x2-4|x|+3 (4)f(x)-x2x(x

1-|x2|1-1(5)f(x)x2x(x

(6)f(x) [g(x)-g(x)](xR)2（2）（3）（4）（5）（6）【解析】(1)∵f(x)的定義域為-1,1，不關于原點對稱，因此f(x)為非奇非偶函數(shù)1-1-1-

-1x1-x21-x2x+2x0且x

x-1,0f(x) (x2)- 1-(-1-f1-(-1--

-f(x，∴f(x) f(-xf(-x)

{g(-x)-g[-(-x)]}

[g(-xg(x)]f(x)，∴f(x

2x2(1)f(x)

x2

(2)f(x)|x1||x1|； (3)f(x)

xx22x1 (x(4)f(x) (x0)x22x1(x（2）（3）（4）【解析】(1)f(xRf(x)

(x)23

3xx23

f(x，f(xf(xRf(x)|x1||x1||x1||x1|f(x，f(xx1f(xx<0，則-x>0f(-x)=-(-x)2+2(-x)+1=-x2-2x+1=-(x2+2x-1)=- 【課堂：函數(shù)的奇偶性356732例2（12（2【變式3】設函數(shù)f(x)和g(x)分別是R上的偶函數(shù)和奇函數(shù)，則下列結論恒成立的是（ A．f(x)+|g(x)|是偶函 B．f(x)-|g(x)|是奇函C．|f(x|+g(x)是偶函數(shù)D．|f(x|-g(x)【答案】2f(x)=x5+ax3-bx-8，f(-2)=10f(2).【答案】-∴8a-2b=- ∴g(-2)=- f(x)+8=x5+ax3-bxg(2)便能迎刃而解.1f(xg(xf(x9g(2)3f(2)為（【答案】g(2f(293,則f(26，f(xf(2)f(26．例3.已知f(x是定義在R上的奇函數(shù)，當x0時f(xx23x1，求f(x的解析式．x23x1,xf(x0xx23x1,x

f(xRf(xf(x)x0x0f(x)f(x)(x)23(x)=x23xf(x在原點有定義，f(0)0x23x1,xf(x)0,xx23x1,x（0，0【課堂：函數(shù)的奇偶性356732例3【變式1】（1）f(x的定義域是Rx0f(xx23x1f(x的解析（2）g(xRx0g(xx22x1g(xx22x1(x 【答案】（1）f(x)x23x1(x0)；（2）g(x) （x x22x1(x4.設定義在[-2，2f(x)在[0，2]f(a1f(a時，求a【答案】2a2 |a1||a 2a1 -2a1-2a

-3a-2a

2a 2 f(xf(xf(|x|)，這樣就減少了討論的麻煩． 【答案】當a=0 a1時，f(x

3aa1時，f(x

a;當 a 時，f(x)

a21

a=0f(x)=x2+|x|+1，此時函數(shù)為偶函數(shù)；a≠0f(x)=x2+|x-a|+1，為非奇非偶函數(shù).xaf(x)(x1)23 ①a②a

1 1時，函數(shù)f(x)在a2

-

且 )2f(x)在axaf(x)x2-xa1(x1)2a a1時，函數(shù)f(x)在-a上單調遞減，f(x)在-，af(a)a22②a

1時，f(x)在-，a上的最小值為f() a，且f()

fa2時，f(x|min4aa2時，f(x|min41a1時，f(x

a21

1f(x|xa||xa|aR【答案】當a0f(x既是奇函數(shù)，又是偶函數(shù)；a0時，函數(shù)f(x)是奇函數(shù)．【解析】對aa0f(x|x||x|0xR，R關于原點對稱，f(xa0

f(x)|xa||xa||xa||xa|f(x)

f(x綜上，當a0f(x既是奇函數(shù)，又是偶函數(shù)；a0時，函數(shù)f(x)是奇函數(shù)．6.yf(x是偶函數(shù)，且在[0，+∞）f(1x2【解析】∵f(x是偶函數(shù)，且在[0，+∞）f(x在（－∞，0]上是增函數(shù)．u=1―x2，