Question

设函数f(x)=1+x分之一

已知函数f(x)=1+x方分之x方 ﹙1﹚求f（2）与f（2分之1）,f（3）与f（3分之1）﹙2﹚

Answer 1

f(x)=x²/(1+x²)
(1)
f(2)=2²/(1+2²)=4/5
f(1/2)=(1/2)²/[1+(1/2)²]=(1/4)/(5/4)=1/5
f(3)=3²/(1+3²)=9/10
f(1/3)=(1/3)²/[1+(1/3)²]=(1/9)/(10/9)=1/10
(2)
f(2)+f(1/2)=4/51/5=1 f(3)+f(1/3)=9/10+1/10=1
猜想：f(x)+f(1/x)=1,为定值.
证：
f(x)+f(1/x)=x²/(1+x²) +(1/x)²/[1+(1/x)²]
=x²/(1+x²)+(1/x²)/(1+ 1/x²)
=x²/(1+x²)+1/(x²+1)
=(x²+1)/(1+x²)
=1