Here is the problem.A nonconstant rational function over the real numbers(a rational function is a function that can be expressed as p ( x ) q ( x ) , with p ( x ), q ( x ) as polynomial function) f ( x ), is defined such that: ( f ( x ) ) 2 − a = f ( x 2 ) for all x ( a is a constant value)Prove that f ( x ) must be of the form x k for some constant k.I have an idea of how to prove it, but I can't do so rigorously. I can prove that any polynomial with more than 2 terms, or monomials, would not work. But I can't prove it won't work for any rational function.

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Here is the problem.A nonconstant rational function over the real numbers(a rational function is a function that can be expressed as             p      (      x      )              q      (      x      )      , with   p  (  x  ),  q  (  x  ) as polynomial function)   f  (  x  ), is defined such that:  (  f  (  x  )      )    2    −  a  =  f  (      x    2    ) for all   x (   a is a constant value)Prove that   f  (  x  ) must be of the form       x    k   for some constant   k.I have an idea of how to prove it, but I can&#039;t do so rigorously. I can prove that any polynomial with more than 2 terms, or monomials, would not work. But I can&#039;t prove it won&#039;t work for any rational function.

Madalynn Acosta · Accepted Answer

If   q  (  z  ) is nonconstant, then   q  (      z    2    ) has at least one complex zero that   q  (  z  ) doesn&#039;t have. (Choose, for example, the zero       z    1   of   q  (      z    2    ) with minimal nonzero argument.) This means that   q  (  z  ) must be constant, for otherwise the right-hand side is undefined at a point where the left-hand side is defined. This reduces to the case where   f is a polynomial, which you have (almost?) solved.Edit: wnoise points out that this argument fails if   q  (  z  ) is a power of   z. Perhaps the argument for polynomials extends to Laurent polynomials (where negative powers of   z are allowed)?

Here is the problem. A nonconstant rational function over the real numbers(a rational function is a

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