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Math Help - f(x+y)=f(x)*f(y), f(1)=q. find the followings.

  1. #1
    jwu
    jwu is offline
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    f(x+y)=f(x)*f(y), f(1)=q. find the followings.

    There are 5 questions on this problem. The one I have difficult with is the last one, I think I got the first four.
    Given a function f(x+y)=f(x)*f(y) and f(1)=q.
    1. Find f(0).
    f(1+0)=f(0)*f(1)=q
    f(0)*q=q
    f(0)=1

    2. express f(-x) in terms of f(x)
    f(-x)=f(x-2x)=f(x)*f(-2x)=f(x)*f(-x)*f(-x)
    1=f(x)*f(-x)
    f(-x)=1/f(x)

    3. find f(n), when n is positive number.
    since f(1)=q
    f(2)=f(1)*f(1)=q^2
    f(3)=f(2)*f(1)=q^3
    f(n)=q^n

    4. find f(n), when n is all integer (Z)
    since f(0)=1=q^0
    f(-1)=1/f(1)=1/q=q^-1
    f(-2)=1/f(2)=1/(q^2)=q^-2
    ...
    f(n)=q^n

    5. find f(n), when n is all rational number.
    I know I need to set n=a/b to take account n is a rational number. But I don't know how to proceed. Looking for help here
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  2. #2
    Senior Member MacstersUndead's Avatar
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    Re: f(x+y)=f(x)*f(y), f(1)=q. find the followings.

    1-4 is correct.

    For 5, you should mention that b has to be non-zero and without loss of generality b>0

    f(a) = f((a/b)*b) = f((a/b) + (a/b) + (a/b) ... + (a/b)) [adding b times] = f(a/b)f(a/b)--f(a/b) [multiplying the function by itself b times] = [f(a/b)]^b

    Hence f(a/b) = [f(a)]^(1/b) = (q^a)^(1/b) [by 4] = q^(a/b) [by exponent rules] which is what we expected.
    Last edited by MacstersUndead; November 20th 2012 at 08:43 PM.
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