Question

已知f(x+1/x)=x²+1/x² +1/x求f(x)

为什么不能《x²+1/x² +1/x》÷(x+1/x)乘x 从而求出f(x)

Answer 1

f[(x+1)/x)]= (x^2+1)/x^2 + 1/x
即 f(1+1/x)=1+1/x^2+1/x = 1+ (1/x)(1+1/x)
= 1+ (1+1/x-1)(1+1/x) ,
令 u=1+1/x
得 f(u) = 1+(u-1)u,
即 f(x)=1+x(x-1)