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Prove-that-f-f-x-is-polynomial-f-x-is-rational-expression-Or-give-a-counter-example-




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

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