Question

若f(x)+f[(x-1)/x]=1+x,求fx

Answer 1

f(x)+f[(x-1)/x]=1+x ==》f(x)=1+x-f[(x-1)/x],(1)(移项得)

==》f[(x-1)/x]=1+(x-1)/x-f{[(x-1)/x-1]/[(x-1)/x]}=1+(x-1)/x-f[-1/(x-1)],(2)(将(1)式中的x换成(x-1)/x得)

f[-1/(x-1)]=1+[-1/(x-1)]-f{[-1/(x-1)-1]/[-1/(x-1)]}=1-1/(x-1)-f(x),(3)(将(1)式中的x换成-1/(x-1)得)

综合(1)(2)(3)得

f[(x-1)/x]=1+(x-1)/x-f[-1/(x-1)]=1+(x-1)/x-[1-1/(x-1)-f(x)]=(x-1)/x+1/(x-1)+f(x),

f(x)=1+x-f[(x-1)/x]=1+x-[(x-1)/x+1/(x-1)+f(x)]=1+x-(x-1)/x-1/(x-1)-f(x)=x+1/x+1/(1-x)-f(x),

2f(x)=x+1/x+1/(1-x),

f(x)=x/2+1/(2x)+1/[2(1-x)],