已知函数f(x)是奇函数,函数φ(x)是偶函数,f(x)+φ(x)=tan(x+π/4),求f(x)和φ(x)的解析式
问题描述:
已知函数f(x)是奇函数,函数φ(x)是偶函数,f(x)+φ(x)=tan(x+π/4),求f(x)和φ(x)的解析式
答
将-x代入,得-f(x)+φ(x)=tan(-x+π/4),然后解二元方程
答
f(x)+φ(x)=tan(x+π/4)
-f(x)+φ(x)=tan(-x+π/4)
φ(x) = (tan(x+π/4) + tan(-x+π/4))/2
f(x) = (tan(x+π/4) - tan(-x+π/4))/2
答
f(x)+g(x)=tan(x+π/4)因为f(x),g(x)分别为奇和偶函数,所以:f(-x)+g(-x)=tan(-x+π/4)可化为:-f(x)+g(x)=tan(-x+π/4)联立这两个式子,解得:g(x)=[tan(x+π/4)+tan(-x+π/4)]/2=f(x)=[tan(x+π/4)-tan(-x+π/4)]/...