已知f(x)=ex-e-x,g(x)=ex+e-x(e=2.71828…)(1)求[f(x)]2-[g(x)]...
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问题详情:
已知f(x)=ex-e-x,g(x)=ex+e-x(e=2.718 28…)
(1)求[f(x)]2-[g(x)]2的值;
(2)若f(x)f(y)=4,g(x)g(y)=8,求的值.
【回答】
解析 (1)[f(x)]2-[g(x)]2=(ex-e-x)2-(ex+e-x)2
=(e2x-2+e-2x)-(e2x+2+e-2x)=-4.
(2)f(x)f(y)=(ex-e-x)(ey-e-y)
=ex+y+e-x-y-ex-y-e-x+y
=[ex+y+e-(x+y)]-[ex-y+e-(x-y)]=g(x+y)-g(x-y)
∴g(x+y)-g(x-y)=4 ①
同理,由g(x)g(y)=8,可得g(x+y)+g(x-y)=8, ②
由①②解得g(x+y)=6,g(x-y)=2,
∴=3.
知识点:基本初等函数I
题型:解答题
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