Question

判斷下列函數(shù)的奇偶性.

(1)f(x)=;

(2)f(x)=x³-2x;

(3)f(x)=a(x∈R);

(4)f(x)=

Answer 1

思路分析:

按奇函數(shù)或偶函數(shù)的定義或幾何特征進(jìn)行判斷即可.

解:(1)函數(shù)的定義域?yàn)閧x|x≠-1},不關(guān)于原點(diǎn)對(duì)稱,所以f(x)既不是奇函數(shù)也不是偶函數(shù).

(2)函數(shù)的定義域?yàn)镽,關(guān)于原點(diǎn)對(duì)稱,f(-x)=(-x)³-2(-x)=2x-x³=-f(x),所以f(x)是奇函數(shù).

(3)函數(shù)的定義域?yàn)镽,關(guān)于原點(diǎn)對(duì)稱,

當(dāng)a=0時(shí),f(x)既是奇函數(shù)又是偶函數(shù);

當(dāng)a≠0時(shí),f(-x)=a=f(x),即f(x)是偶函數(shù).

(4)函數(shù)的定義域?yàn)镽,關(guān)于原點(diǎn)對(duì)稱,

當(dāng)x≥0時(shí),-x＜0,此時(shí)f(-x)=-x[1+(-x)]=-x(1-x)=-f(x);

當(dāng)x＜0時(shí),-x≥0,此時(shí)f(-x)=-x[1+(-x)]=-x(1-x)= -f(x);

當(dāng)x=0時(shí),-x＜0,此時(shí)f(-x)=0,f(x)=0,即f(-x)=-f(x);

綜上,f(-x)=-f(x),所以f(x)為奇函數(shù).