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ISI2017-MMA-24

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The number of polynomial function $f$ of degree $\geq$ 1 satisfying $f(x^{2})=(f(x))^{2}=f(f(x))$ for all real $x$, is

  1. $0$
  2. $1$
  3. $2$
  4. infinitely many
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ANS:B(one polynomial)

The only possible polynomial f(x) is (square of x) i.e;f(x)=x^2

Let f(x)=x^2 => f(x)^2=x^4

f(x^2)=(x^2)^2=x^4

f(f(x))=f(x^2)=x^4

Therefore f(x^2)=f(x)^2=f(f(x))
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