1f(x) =0 is a trivial solution
f(x) = x/(1+ax) is the non-trivial solution
\hspace{-16}$Find a function $\mathbf{f(x)\neq x}$ such that for every $\mathbf{x\geq 0}$\\\\ $\mathbf{f\left(\frac{x}{1+x}\right)=\frac{f(x)}{1+f(x)}}$\\\\
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4 Answers
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341We have
f \left(\frac{1}{1+x} \right) = \frac{f\left(\frac{1}{x} \right)}{1+f\left(\frac{1}{x} \right)}
\Rightarrow \frac{1}{f \left(\frac{1}{1+x} \right)} = \frac {1}{f\left(\frac{1}{x} \right)}+1
So if g(x) = \frac {1}{f\left(\frac{1}{x} \right)}
we have g(x+1) = g(x)+1 \Leftrightarrow g(x) = x+k
Hence
f\left(\frac{1}{x} \right) = \frac{1}{ x+k}
and
f(x)= \frac{x}{ 1+kx}
341there is a small snag here. will sort it out as soon as i am back
