Question

若(x-1)f(x+1/x-1)+f(x)=x 求f（x）

Answer 1

(x-1)f(x+1/x-1)+f(x)=x (1)
用(x+1)/(x-1)代替上式中的x得
[(x+1)/(x-1)-1]f{[(x+1)/(x-1)+1]/[(x+1)/(x-1)-1]}+f[(x+1)/(x-1)]=(x+1)/(x-1)
2f(x)+f[(x+1)/(x-1)]=(x+1)/(x-1)
2(x-1)f(x)+(x-1)f[(x+1)/(x-1)]=x+1 (2)
(2)-(1)得(2x-3)f(x)=1
f(x)=1/(2x-3)

Answer 2

设t=(x+1)/(x-1),则tx-t=x+1,(t-1)x=t+1,x=(t+1)/(t-1),x-1=2/(t-1),
由(x-1)f[(x+1)/(x-1)]+f(x)=x,(1)
得2/（x-1)f(x)+f[(x+1)/(x-1)]=(x+1)/(x-1),(2)
(2)*(x-1)-(1),得
f(x)=1.

Answer 3

用x+1/x-1替代上式中的x得
2/(x-1)f(x)+f(x+1/x-1)=x+1/x-1
与上式联立可解出
f(x)=1为常数函数

Answer 4

f(x)=1 ??