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Thread: Extracurricular Elementary Mathematics Marathon

  1. #76
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by seanieg89 View Post
    I do not believe your first solution is correct, neither is the statement that if f(x) is a solution then f(2x/(3x-2)) is one too.

    This would be the case if the functional equation was symmetric in these two unknowns, but the factor of 1/2 stops that from happening.

    A quick way to verify that this solution doesn't work without plugging into the functional equation directly is by considering the limit L of f(x)/x as x->0. The functional equation implies that L must be 1008 if it exists, whereas the limit in your first proposed solution is -1008.

    The second solution is the unique solution and you are done after solving the simultaneous equations. (Because all of the previous lines are quantified over all x, you can actually be sure that this is the unique solution after reaching the line f(x)=blah).


    (I obviously cooked up the numbers so that 2016 featured in the unique solution.)
    Ah. Knew I should have just done a simple substitution instead of the involution property.

    Was that intentional, by the way?
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by Paradoxica View Post
    Ah. Knew I should have just done a simple substitution instead of the involution property.

    Was that intentional, by the way?
    Yep, you got the key idea, just a silly error.

    Was what intentional?

  3. #78
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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by seanieg89 View Post
    Yep, you got the key idea, just a silly error.

    Was what intentional?
    Involution
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by Paradoxica View Post
    Involution
    Oh yes, entirely. That was why I considered it cute, I quite like simple ideas exploiting symmetries like that.

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    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon

    Find with proof, all continuous functions f:R->R such that:

    f(f(f(x)))=x for all real x.

  6. #81
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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by seanieg89 View Post
    Find with proof, all continuous functions f:R->R such that:

    f(f(f(x)))=x for all real x.
    I'm doubt the validity of my solution, and I'm certain there are massive logical gaps, but here's what I have:

    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by Paradoxica View Post
    I'm doubt the validity of my solution, and I'm certain there are massive logical gaps, but here's what I have:

    Yeah, you have done something fishy in deducing that f=f^{-1}, care to explain your reasoning if you still believe this fact after thinking more?

    Note also that you have used continuity nowhere. This is essential, as we have a vast array of solutions to the functional equation if continuity is not required:

    Partition the reals into an uncountable union of sets, each with either 1 or 3 elements.
    Define f to map elements of singleton sets to themselves and to cycle the three elements in each of the other sets.

    Any such function f will satisfy the functional equation, but almost all of them will be highly discontinuous.

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    Re: Extracurricular Elementary Mathematics Marathon

    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by Paradoxica View Post
    Composition cannot create parity? Why not? Eg f(x)=1-x is not odd or even, but composed with itself is the identity which is odd.

    It is true that the only polynomial function that works is the identity and yes degree considerations give you this quickly as you note.

    This isn't too fruitful a way of thinking about what general continuous functions can solve the equation though.

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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by seanieg89 View Post
    Composition cannot create parity? Why not? Eg f(x)=1-x is not odd or even, but composed with itself is the identity which is odd.

    It is true that the only polynomial function that works is the identity and yes degree considerations give you this quickly as you note.

    This isn't too fruitful a way of thinking about what general continuous functions can solve the equation though.
    No discontinuities allowed, this includes singularities of rational functions? for example, 1/(1-x) is clearly a solution to the functional equation, but has a simple pole at x=1
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  11. #86
    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Elementary Mathematics Marathon

    Quote Originally Posted by Paradoxica View Post
    No discontinuities allowed, this includes singularities of rational functions? for example, 1/(1-x) is clearly a solution to the functional equation, but has a simple pole at x=1
    No, singularities are not allowed, because the function is continuous and has domain R. (We cannot have any domain "holes".)

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    Re: Extracurricular Elementary Mathematics Marathon



    kawaiipotato likes this.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  13. #88
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    Re: Extracurricular Elementary Mathematics Marathon

    Last edited by Paradoxica; 12 Mar 2016 at 12:48 AM.
    KingOfActing likes this.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  14. #89
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    Re: Extracurricular Elementary Mathematics Marathon

    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  15. #90
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    Re: Extracurricular Elementary Mathematics Marathon

    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  16. #91
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    Re: Extracurricular Elementary Mathematics Marathon

    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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