f(x)=a^x-a^(-x)
=a^x-1/a^x
=(a^2x-1)/a^x
a^2x-1=a^xf(x)
a^2x-f(x)a^x=1
a^2x-f(x)a^x+f^2(x)/4=1+f^2(x)/4=(4+f^2(x))/4
(a^x+f(x)/2)^2=(4+f^2(x))/4
a^x+f(x)/2=±√(4+f^2(x)) /2
a^x=-f(x)/2±√(4+f^2(x)) /2=1/2*(f(x)±√(4+f^2(x))
lna^x=xlna=ln[1/2*(f(x)±√(4+f^2(x))]
x=ln[1/2*(f(x)±√(4+f^2(x))]/lna
=loga [1/2*(f(x)±√(4+f^2(x))]
所以它的反函數(shù)是
f(x)= loga [1/2*(x±√(4+x^2)]a^2x-f(x)a^x+f^2(x)/4=1+f^2(x)/4=(4+f^2(x))/4這一步是怎么出來的呢？就是對一元二次方程配方如ax^2+bx+c配成a(x+b/2a)^2+(4ac-b^2)/4a