函数f(x)在(－∞ ＋∞)内有定义 f(x)不恒等于1 下列给出的函数哪些必为奇函数？哪些必为偶函

正确答案：(1)设g(x)＝f(x²)则g(－x)＝f((－x)²)＝f(x²)＝g(x)∴f(x²)必为偶函数.(2)设g(x)＝xf(x²)则g(－x)＝(－x)f[(－x)²＝－xf(x²)＝－g(x)∴xf(x²)必为奇函数.(3)设g(x)＝x²f(x)则g(－x)＝(－x)²f(－x)＝x²f(－x)∵f(－x)奇偶性不能确定∴x²f(x)奇偶性不定.(4)设g(x)＝f²(x)则g(－x)＝f²(－x)≠f²(x)且f²(x)≠－f²(x)∴f²(x)奇偶性不定.(5)设g(x)＝f(｜x｜)则g(－x)＝f(｜x｜)＝f(｜x｜)＝g(x)∴f(｜ x ｜)必为偶函数．(6)设g(x)＝｜f(x)｜则g(－x)＝｜ f(－x)｜≠｜f(x)｜且｜f(－x)｜≠－｜f(x)｜∴｜(x)｜奇偶性不定．(7)设g(x)=f(x)＋f(－x)则g(－x)＝f(－x)＋f[－(－x)＝f(－x)＋f(x)＝g(x)∴f(x)＋f(－x)必偶函数．(8)设g(x)＝f(x)－f(－x)则g(－x)＝f(－x)－f[－(-x)＝f(－x)－f(x)＝[f(x)－f(－(x))＝－g(x)∴f(x)－f(－x)必为奇函数．
(1)设g(x)＝f(x2),则g(－x)＝f((－x)2)＝f(x2)＝g(x)∴f(x2)必为偶函数.(2)设g(x)＝xf(x2),则g(－x)＝(－x)f[(－x)2＝－xf(x2)＝－g(x)∴xf(x2)必为奇函数.(3)设g(x)＝x2f(x),则g(－x)＝(－x)2f(－x)＝x2f(－x)∵f(－x)奇偶性不能确定∴x2f(x)奇偶性不定.(4)设g(x)＝f2(x),则g(－x)＝f2(－x)≠f2(x)且f2(x)≠－f2(x)∴f2(x)奇偶性不定.(5)设g(x)＝f(｜x｜),则g(－x)＝f(｜x｜)＝f(｜x｜)＝g(x)∴f(｜x｜)必为偶函数．(6)设g(x)＝｜f(x)｜,则g(－x)＝｜f(－x)｜≠｜f(x)｜且｜f(－x)｜≠－｜f(x)｜∴｜(x)｜奇偶性不定．(7)设g(x)=f(x)＋f(－x),则g(－x)＝f(－x)＋f[－(－x)＝f(－x)＋f(x)＝g(x)∴f(x)＋f(－x)必偶函数．(8)设g(x)＝f(x)－f(－x),则g(－x)＝f(－x)－f[－(-x)＝f(－x)－f(x)＝[f(x)－f(－(x))＝－g(x)∴f(x)－f(－x)必为奇函数．