试题与答案

我们给出如下定义:对函数y=f(x),x∈D,若存在常数C(C∈R),对任意的x

题型:解答题

题目:

我们给出如下定义:对函数y=f(x),x∈D,若存在常数C(C∈R),对任意的x1∈D,存在唯一的x2∈D,使得
f(x1)+f(x2)
2
=C
,则称函数f(x)为“和谐函数”,称常数C为函数f(x)的“和谐数”.
(1)判断函数f(x)=x+1,x∈[-1,3]是否为“和谐函数”?答:______.(填“是”或“否”)如果是,写出它的一个“和谐数”:______.(4分)
(2)证明:函数g(x)=lgx,x∈[10,100]为“和谐函数”,
3
2
是其“和谐数”;
(3)判断函数u(x)=x2,x∈R是否为和谐函数,并作出证明.

答案:

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