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Adequate proof?

  1. Dec 10, 2016 #1
    1. The problem statement, all variables and given/known data
    I have been going through a textbook trying to solve some of of these with somewhat formal proofs. This is a former Putnam exam question. (Seemingly the easiest one I have attempted, which worries me).

    Consider a polynomial function ƒ with real coefficients having the property ƒ(g(x)) = g(ƒ(x)) for all polynomial functions g with real coefficients. Determine and prove the nature of ƒ.

    2. Relevant equations
    No equations, just the tricky fact that constant functions are polynomials.

    3. The attempt at a solution
    A proof by contradiction that ƒ(x) is the identity function:

    We can consider the function h(x) = 1 and its derivative, which I will denote as g, namely to state the fact that ƒ((g+h)(x)) = ƒ(1) = g(ƒ(x)) + h(ƒ(x)), which then = ƒ(0) + ƒ(1); therefore ƒ(0) = 0.
    Now, suppose that ƒ is not the identity function, or, specifically, that ƒ(c) ≠ c. We can consider the polynomial function k(x) which has a zero at c but not at ƒ(c). Then ƒ(k(c)) = ƒ(0) = 0 ≠ k(ƒ(c)); a contradiction occurs. This proves that ƒ must be the identity function.
     
  2. jcsd
  3. Dec 10, 2016 #2

    PeroK

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    I can't see what you are doing in the first part. There is a very simple way to show that ##f(0) = 0##.

    The second part is sound. Although, this could be simplified by doing a straight proof, as opposed to using contradiction.
     
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