设fn(x)=f{f[...f(x)]},f(x)=x/√1+x^2 求fn(x)我先猜想，再用数学归纳法证明出猜想，定义域怎么求？
2019-04-14
设fn(x)=f{f[...f(x)]},f(x)=x/√1+x^2 求fn(x)
我先猜想，再用数学归纳法证明出猜想，定义域怎么求？
优质解答
f1(x)=x/√1+x^2=√[(x^2)/(1+x^2)]
设fn(x)=√[(x^2*t)/(1+x^2*t)]
则fn+1(x)=f(fn(x))=√[x^2*(2t)/(1+x^2*(2t)]
故fn(x)=√[(x^2*2^(n-1))/(1+x^2*2^(n-1))]
f1(x)=x/√1+x^2=√[(x^2)/(1+x^2)]
设fn(x)=√[(x^2*t)/(1+x^2*t)]
则fn+1(x)=f(fn(x))=√[x^2*(2t)/(1+x^2*(2t)]
故fn(x)=√[(x^2*2^(n-1))/(1+x^2*2^(n-1))]