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

P r o v e t h a t

^{″} f (f (x)) i s p o l y n o m i a l^{″} \Rightarrow^{″} f (x) i s r a t i o n a l e x p r e s s i o n^{″}

O r g i v e a c o u n t e r e x a m p l e .

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 .