Question

设函数f(x)=2^x-a/2^x是偶函数,则常数a=

Answer 1

因为f(x)为偶函数,由偶函数的概念知,f(x)=f(-x) 恒成立.

又 f(x) = 2^x-a/2^x,--------------(1)

所以,f(-x) = 2^(-x) - a/2^(-x) = 1/2^x - a*2^x,--------------(2)

所以,2^x-a/2^x = 1/2^x - a*2^x,--------------(3)

整理(3)式得,(1 - 2^(2*x)) (1+ a) = 0 恒成立;

由于1 - 2^(2*x)) 不恒为0,

所以1+ a = 0,即a = -1.