Question 41949
{{{f(x) = c*x/ (2x + 3) }}}
Therefore, {{{f(f(x)) = c*f(x)/ (2*f(x) + 3) }}}


[Just replace 'x' with f(x)]


Thus, {{{f(f(x)) = c*(c*x/(2x + 3))/(2*(c*x/ (2x + 3)) + 3)}}}
or {{{f(f(x)) = (c^2*x/(2x + 3))/((2*c*x+6x+9)/ (2x + 3))}}}
or {{{f(f(x)) = c^2*x/(2*c*x+6x+9)}}}


So, if f(f(x)) = x then
{{{c^2*x/(2*c*x+6x+9) = x}}}
or {{{c^2/(2*c*x+6x+9) = 1}}}
or {{{c^2 = 2*c*x+6x+9}}}
or {{{c^2-9=2cx+6x}}}
or {{{x=(c^2-9)/(2c+6)}}}


As x cannot be equal to {{{-3/2}}} 
so {{{(c^2-9)/(2c+6)}}} can also not be equal to {{{-3/2}}}.


But, however, if {{{(c^2-9)/(2c+6) = -3/2}}} then
{{{2(c^2-9)+3(2c+6) = 0}}}
or {{{2c^2+6c = 0}}}
or {{{c(c+3)=0}}}
Hence either c = 0 or c = -3.
Thus f(f(x)) = x is valid for all values of 'x' except c = 0 and c = -3.