设f(x+1/x)=x2+1/x2,求f(x)设u=x+1/x,求f(u)=(u^2±√u^2-4)^2/4+4/(u^2±√u^2-4)^2=u^2-2过程

Question

设f(x+1/x)=x2+1/x2,求f(x)
设u=x+1/x,求f(u)=(u^2±√u^2-4)^2/4+4/(u^2±√u^2-4)^2=u^2-2过程

Answer 1

设x+1/x=u
则(x+1/x)²=u²
x²+1/x²+2=u²
x²+1/x²=u²-2
∴f(u)=u²-2
即f(x)=x²-2
由u=x+1/x中解出x=u^2±√(u^2-4)的解法特笨,我上面的解法简单明了好懂.