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

2f[(x-1)/x]+f[(x+1)/x]=1+x
2f(1-1/x)+f(1+1/x)=1+x……(1)
令x=-x(意思是，不用写在步骤中)
2f(1+1/x)+f(1-1/x)=1-x……(2)
（2）*2－（1）,得：
f(1+1/x)=1/3-x
换元法：t=1+1/x,x=1/(t-1)
f(t)=1/3-1/(t-1)
f(x)=1/3-1/(x-1),x不为1

Answer 3