Question

f(x+1/x)=(x+x^3)/(1+x^4) 求f(x)

Answer 1

f(x+1/x)=(x+x^3)/(1+x^4)
设x+1/x=u
(x^2+1)/x=u
x^2-ux+1=0
这样是够复杂的，你会不会把(x+1)/x打成x+1/x了

我宁愿认为你是打错了，否则那根本没法解啊
设u=(x+1)/x
ux=x+1
x(u-1)=1
x=1/(u-1)
f(u)=[1/(u-1)+1/(u-1)^3]/[1+1/(u-1)^4]={[(u-1)^2+1]/(u-1)^3}/{[(u-1)^4+1]/(u-1)^4}
=(u-1)[(u-1)^2+1]/[(u-1)^4+1]=[(u-1)^3+u-1]/[(u-1)^4+1]=(u^2-2u+2)(u-1)/[(u-1)^4+1]
所以f(x)=(x^2-2x+2)(x-1)/[(x-1)^4+1]

Answer 2

解：f(x+1/x)=(x+x^3)/(1+x^4)
=（1/x+x)/(1/x^2+x^2) (分子分母同除以x^2)
=（1/x+x)/[(1/x^2+x^2)+2-2]
=（1/x+x)/[(1/x+x)^2-2]
令x+1/x=X
得f(x+1/x)=(x+x^3)/(1+x^4)
=F(X)=X/(X^2-2)

Answer 3

f(x+1/x)=(x+x^3)/(1+x^4)，分子分母同时除以x^2,得f(x+1/x)=(1/x+x)/(1/x^2+x^2)=)=(1/x+x)/[(x+1/x)^2-2],所以f(x)=x/(x^2-2)