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Math Help - Arbitrary powers

  1. #1
    Member akhayoon's Avatar
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    Arbitrary powers

    the question states

    f(x)=(1/2) (a^x+a^-x) if a>0. If f(x+y)+f(x-y)=kf(x)f(y), where k is a constant find that constant k

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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by akhayoon View Post
    the question states

    f(x)=(1/2) (a^x+a^-x) if a>0. If f(x+y)+f(x-y)=kf(x)f(y), where k is a constant find that constant k

    did you write out the equation to see if you got any ideas? what have you tried?

    note that:

    f(x) = \frac 12 \left( a^x + a^{-x} \right)

    f(y) = \frac 12 \left( a^y + a^{-y} \right)

    f(x + y) = \frac 12 \left( a^{x + y} + a^{-x - y} \right)

    f(x - y) = \frac 12 \left( a^{x - y} + a^{y - x} \right)

    now put those together as the equation suggests and see what you get
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    Quote Originally Posted by akhayoon View Post
    the question states

    f(x)=(1/2) (a^x+a^-x) if a>0. If f(x+y)+f(x-y)=kf(x)f(y), where k is a constant find that constant k

    This is a bit of generalised triganometry

    Suppose for some a that:

    f(x)=(1/2) (x^x+a^{-x})

    Then:

     <br />
f(x+y)+f(x-y) = \frac{1}{2}\left[ a^{x+y}+a^{-x-y}\right]+\frac{1}{2}\left[ a^{-x-y}+a^{-x+y}\right]

    .................. =\frac{1}{2} a^x \left[ a^{y}+a^{-y}\right]+\frac{1}{2} a^{-x} \left[ a^{y}+a^{-y}\right]=2f(x)f(y)

    ZB
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Constatine11 View Post
    Suppose for some a that:

    f(x)=(1/2) (x^x+a^{-x})

    Then:

     <br />
f(x+y)+f(x-y) = \frac{1}{2}\left[ a^{x+y}+a^{-x-y}\right]+\frac{1}{2}\left[ a^{-x-y}+a^{-x+y}\right]

    .................. =\frac{1}{2} a^x \left[ a^{y}+a^{-y}\right]+\frac{1}{2} a^x \left[ a^{y}+a^{-y}\right]
    where did that quote in your signature come from?
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    Quote Originally Posted by Jhevon View Post
    where did that quote in your signature come from?
    The quote comes from Isaac Newton the "giants" he is referring to are the mathematicians and astronomers of the past: Kepler, Galileo, probably Archimedes, but the most important giant that inspired Newton the most was Fermat.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    The quote comes from Isaac Newton the "giants" he is referring to are the mathematicians and astronomers of the past: Kepler, Galileo, probably Archimedes, but the most important giant that inspired Newton the most was Fermat.
    Nice. it's a great quote
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  7. #7
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    Quote Originally Posted by Jhevon View Post
    Nice. it's a great quote
    How about:
    "And perhaps posterity shall thank me for having shown the Ancients did not know everything" (Fermat).

    I am not exactly sure what I means, but it sounds like a King James type of quotation which are really eqoulently stated. I wish I could talk like that (in old English) but I am not so good.
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