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Math Help - Functional equation

  1. #1
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    Functional equation

    Hi, I need help on these two problems dealing with functional equation

    Let f be a function defined everywhere on the real axis. Suppose also that f satisfies the functional equation

    f(x+y) = f(x)f(y) for all x and y.

    A) Using only the functional equation, prove that f(0) is either 0 or 1. Also, prove that if f(0) then f(x) is not equal to 0 for all x.

    Thanks
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by BrianW View Post
    Hi, I need help on these two problems dealing with functional equation

    Let f be a function defined everywhere on the real axis. Suppose also that f satisfies the functional equation

    f(x+y) = f(x)f(y) for all x and y.

    A) Using only the functional equation, prove that f(0) is either 0 or 1.
    f(0) = f(0+0) = f(0)*f(0) = [f(0)]^2

    so we have f(0) = [f(0)]^2
    => [f(0)]^2 - f(0) = 0
    => f(0)[f(0) - 1] = 0
    => f(0) = 0 or f(0) - 1 = 0
    => f(0) = 0 or f(0) = 1

    Also, prove that if f(0) then f(x) is not equal to 0 for all x.
    i believe a part of this question is missing. "if f(0) is what? then f(x) is not equal to 0 for all x."
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  3. #3
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    oh sorry, I forgot that part.

    It should be: "Also, prove that if f(0) is not equal to 0, then f(x) is not equal to 0 for all x"
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by BrianW View Post
    oh sorry, I forgot that part.

    It should be: "Also, prove that if f(0) is not equal to 0, then f(x) is not equal to 0 for all x"
    From above, we see if f(0) is not zero, then f(0) = 1

    now recall f(x + y) = f(x)f(y)
    => f(x) = f(x + y)/f(y)
    if f(x + y) = f(0) we have:
    f(x) = f(0)/f(y)
    => f(x) = 1/f(y) if f(0) is not 0
    but 1/f(y) is never zero. thus we have f(x) is never zero for any x
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  5. #5
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    Quote Originally Posted by BrianW View Post
    oh sorry, I forgot that part.

    It should be: "Also, prove that if f(0) is not equal to 0, then f(x) is not equal to 0 for all x"
    I do not like what Jhevon did because he did not show that f(y)!=0.

    ---

    We are given that f(0) is non-zero.
    Note, for any real x we have that,
    f(0) = f[(x)+(-x)] =f(x)*f(-x)
    Now since the left hand side is non-zero implies f(x) is non-zero.
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  6. #6
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    I do not like what Jhevon did because he did not show that f(y)!=0.
    i did not think that was necessary (f(x) won't be zero no matter what f(y) is). but ok, i guess your way leaves no uncertainty
    Last edited by Jhevon; May 20th 2007 at 10:15 PM.
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