设f(x)在(－∞,＋∞)内有定义,证明:f(x)+f(－x)为偶函数,而f(x)-f(－x)为奇函数.
\u8bbe\u51fd\u6570f(x)\u5b9a\u4e49\u5728(-l\uff0cl)\u4e0a\uff0c\u8bc1\u660ef(x)+f(-x)\u662f\u5076\u51fd\u6570\uff0cf(x)+f(-x)\u662f\u5947\u51fd\u6570\u3002\u4ee4g(x) = f(x)+f(-x),-l<x<l\uff0c
\u663e\u7136-l<-x<l
\u5219g(-x) = f(-x)+f(-(-x)) = f(-x)+f(x) = g(x)
\u6240\u4ee5g(x)\u4e3a(-l,l)\u4e0a\u7684\u5076\u51fd\u6570
\u4ee4g(x) = f(x)-f(-x),-l<x<l\uff0c
\u663e\u7136-l<-x<l
\u5219g(-x) = f(-x)-f(-(-x)) = f(-x)-f(x) =- g(x)
\u6240\u4ee5g(x)\u4e3a(-l,l)\u4e0a\u7684\u5947\u51fd\u6570
\uff08-L,L)\u533a\u95f4\u5bf9\u79f0
\u8bbet(x)=F\uff08x\uff09+F\uff08-x\uff09
\u90a3\u4e48t(-x)=F\uff08-x\uff09+F\uff08x\uff09=t(x)
\u6240\u4ee5\u5076\u51fd\u6570
\u8bbeg(x)=F\uff08x\uff09-F\uff08-x\uff09
\u90a3\u4e48g(-x)=F\uff08-x\uff09-F\uff08x\uff09=-\uff08-F\uff08-x\uff09+F\uff08x\uff09\uff09=-g(x)
\u6240\u4ee5\u5947\u51fd\u6570
\u4e0d\u719f\u7684\u8bdd\u5c31\u591a\u5199t(x)=F\uff08x\uff09+F\uff08-x\uff09\u548cg(x)=F\uff08x\uff09-F\uff08-x\uff09\u8fd9\u4e00\u6b65\u5c31\u53ef\u4ee5\u4e86
则 任取x∈(－∞,＋∞),
g(-x)=f(-x)+f[-(-x)]=g(x)
h(-x)=f(-x)-f[-(-x)]=f(-x)-f(x)=-h(x)
所以 g(x)是偶函数,h(x)是奇函数。
用手机不好打符号，用定义法。爱符负爱可司等于爱符爱可司就是偶爱符符爱可司等于符爱符爱可司为奇
（1）令T(x)=f(x)+(-x).则T(-x)=f(-x)+f(x),因为定义域为R,所以T(-x)=T(x)，所以此函数为偶函数（2）令B(x)=f(x)_f(-x),则B(-x)=f(-x)_f(x),因为定义域为R,所以B(x)=-B(x),所以此函数为奇函数
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