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Mapping Functions

  1. Feb 22, 2012 #1
    1. The problem statement, all variables and given/known data
    Let F be the set of all functions f mapping ℝ into ℝ that have derivatives of all orders. Determine whether the given map ∅ is an isomorphism of the first binary structure with the second.

    {F,+} with {ℝ,+} where ∅(f)=f'(0)

    2. Relevant equations

    None

    3. The attempt at a solution
    If I have to determine it is isomorphism, I have to prove it is onto and 1-1, but I'm not sure how to do that with ∅(f)=f'(0)
     
  2. jcsd
  3. Feb 22, 2012 #2
    Is ∅ injective? (Same as asking if it's one-to-one). That is, does ∅ always send different functions to different reals?
     
  4. Feb 22, 2012 #3

    Deveno

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    i suggest taking a handful of functions, and calculating f'(0) for each of them.

    possible candidates:

    f(x) = xn (don't forget the special cases n = 0, and n = 1)
    f(x) = ax + b
    f(x) = sin(x)
    f(x) = cos(x)

    do any of these have the property that θ(f1) = θ(f2) but f1 ≠ f2?
     
  5. Feb 22, 2012 #4
    so I would try doing f(x)=X^n. and then get 0^0 and 0^1? so the answers would be 1 and 0. I'm still a little bit confused
     
  6. Feb 22, 2012 #5

    Dick

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    If that's confusing you, start with f(x)=x and f(x)=sin(x).
     
  7. Feb 22, 2012 #6

    Mark44

    Staff: Mentor

    If n = 0, then f(x) = 1, so f(0) = 1.
    If n = 1, then f(x) = x, so f(0) = 0.
     
  8. Feb 22, 2012 #7

    Deveno

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    unless i am mistaken, the problem is asking for the derivative of f at 0, not f(0).
     
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