Question

设f（x）=(1+x^2)/(1-x^2) 。求证:f(1/x)=-f(x)

请看清。(1+x^2)/(1-x^2) 中【x^2】

Answer 1

证明：∵f(x)=(1+x²)/(1-x²)
∴f(1/x)=(1+1/x²)/(1-1/x²)
=[(x²+1)/x²]/[(x²-1)/x²]
=(x²+1)/(x²-1)
=-[(1+x²)/(1-x²)]
=-f(x)
即左边=右边
故原命题成立。

Answer 2

f(1/x)=（1+(1/x)^2）/(1-(1/x)^2)
=(1+1/x^2) /(1-1/x^2)
=(x^2+1)/x^2 /(x^2-1)/x^2 (消掉x^2 ）
=(1+x^2)/(x^2-1)
=-f(x)

Answer 3

证明:：∵f(x)=(1+x²)/(1-x²)
∴f(1/x)=(1+1/x²)/(1-1/x²)
=[(x²+1)/x²]/[(x²-1)/x²]
=(x²+1)/(x²-1)
=-[(1+x²)/(1-x²)]
=-f(x)
∴f(1/x)=-f(x)

Answer 4

f(1/x)=(1/(1/x)^2)/(1-(1/x^2))=(x^2+1)/(x^2-1)=-(1+x^2)/(1-x^2)=-f(x)

Answer 5

f(1/x)=（1+1/x²)/(1-1/x²)=(1+x²)(x²-1)=-(1+x²)(1-x²)=-f(x)