Question Number 1289 by Rasheed Soomro last updated on 19/Jul/15
$${Prove}\:{that}\: \\ $$$$\:\:''\:{f}\left(\:{f}\left({x}\right)\:\right)\:{is}\:{polynomial}\:''\:\Rightarrow\:''\:{f}\left({x}\right)\:{is}\:{rational}\:{expression}\:'' \\ $$$$\:{Or}\:{give}\:{a}\:{counter}\:{example}\:. \\ $$$$ \\ $$
Answered by prakash jain last updated on 20/Jul/15
$$\mathrm{Counter}\:\mathrm{Example} \\ $$$${g}\left({x}\right)=\left({a}−{x}^{\mathrm{2}} \right)^{\mathrm{1}/\mathrm{2}} \\ $$$${g}\left({g}\left({x}\right)\right)=\left({a}−{g}\left({x}\right)^{\mathrm{2}} \right)^{\mathrm{1}/\mathrm{2}} \\ $$$$\left({g}\left({x}\right)\right)^{\mathrm{2}} ={a}−{x}^{\mathrm{2}} \\ $$$${a}−\left({g}\left({x}\right)\right)^{\mathrm{2}} ={x}^{\mathrm{2}} \\ $$$${g}\left({g}\left({x}\right)\right)={x}\:\left(\mathrm{polynomial}\right) \\ $$$$\mathrm{This}\:\mathrm{is}\:\mathrm{another}\:\mathrm{solution}\:\mathrm{to}\:{f}\left({f}\left({x}\right)\right)={x} \\ $$$$\mathrm{Many}\:\mathrm{more}\:\mathrm{counter}\:\mathrm{examples}\:\mathrm{can}\:\mathrm{be}\:\mathrm{created}. \\ $$