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# If $f(X)=\large\frac{kx}{1+x}$, $x\neq1$ and $fof(x)=x$, then find $k$.

$\begin{array}{1 1} \sqrt 2 \\ - \sqrt 2 \\ -1 \\ 1 \end{array}$

Can you answer this question?

Given $f(f(x))=\large\frac{k.f(x)}{1+f(x)}$ $=x$
$\Rightarrow\:\large\frac{k.\large\frac{kx}{1+x}}{1+\large\frac{kx}{1+x}}$ $=x$
$\Rightarrow\:\large\frac{k^2x}{1+x+kx}$ $=x$
$\Rightarrow\:(k+1)x^2+x(1-k^2)=0$
comparing the coefficient of $x^2\:and\:x$ on either side we get
$k+1=0$ and $1-k^2=0$
$\Rightarrow\: k=-1$
answered May 25, 2013