Results 1 to 5 of 5

Math Help - f(x).f(1/x)

  1. #1
    Newbie
    Joined
    Mar 2009
    Posts
    16

    f(x).f(1/x)

    If f(x).f(1/x) = f(x) + f(1/x) and f(4) = 65,

    what will be the value of f(6) ?

    Any help would be greatly appreciated.

    Thanks,
    Anshu
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Spec's Avatar
    Joined
    Aug 2007
    Posts
    318
    \left\{\begin{array}{ll}f(x)f\left(\frac{1}{x}\rig  ht)=f(x)+f\left(\frac{1}{x}\right)\\f(4)=65\end{ar  ray}\right. \Longleftrightarrow 65f\left(\frac{1}{4}\right)=65+f\left(\frac{1}{4}\  right)\Longleftrightarrow f\left(\frac{1}{4}\right)=\frac{65}{64} (1)

    Now if f is a homogeneous function, then f(\alpha x)=\alpha f(x). Thus f(6)=f\left(\frac{24}{4}\right)=24f\left(\frac{1}{  4}\right) (2)

    Now use (1) and (2).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Curious_eager View Post
    If f(x).f(1/x) = f(x) + f(1/x) and f(4) = 65,

    what will be the value of f(6) ?
    The answer could be almost anything, since the given information only determines the values of the function at x=4 and x=1/4. Probably the simplest function to satisfy the functional equation is f(x) = 1+x^3, which would make f(6)=217.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member Spec's Avatar
    Joined
    Aug 2007
    Posts
    318
    If it's a homogeneous function you don't even need to go through all the steps that I showed since you can just use the fact that f(4)=65 and f(\alpha\cdot 4)=\alpha \cdot 65

    Try posting the question exactly as it was given.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member pankaj's Avatar
    Joined
    Jul 2008
    From
    New Delhi(India)
    Posts
    317
    The question actually is that :

    If f(x) is a polynomial function satisfying the condition that

    f(x)f(\frac{1}{x})=f(x)+f(\frac{1}{x})

    and f(4)=65.Then what is value of f(6).


    On assuming a poloynomial of degree n and plugging it in the given condition and equating the coefficients we get

     <br />
f(x)=1\pm x^n<br />

    Consequently, f(x)=1+x^3
    Follow Math Help Forum on Facebook and Google+

Search Tags


/mathhelpforum @mathhelpforum