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Math Help - Function

  1. #1
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    Function

    Find all functions f: \mathbb{N^{*}} \rightarrow \mathbb{N^{*}} which satisfy the following conditions
    i) f is strictly increasing;
    ii) f(f(n))= 4n+9 for all n \in \mathbb{N^{*}};
    iii) f(f(n)-n)=2n+9 for all n \in \mathbb{N^{*}}.
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  2. #2
    Newbie Sorombo's Avatar
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    I found one, but only by trial and errors.

    f(n)=2n+3

    f(2n+3)=2(2n+3)+3=4n+9

    f(2n+3-n)=2(n+3)+3=2n+9
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by unlimited View Post
    Find all functions f: \mathbb{N^{*}} \rightarrow \mathbb{N^{*}} which satisfy the following conditions
    i) f is strictly increasing;
    ii) f(f(n))= 4n+9 for all n \in \mathbb{N^{*}};
    iii) f(f(n)-n)=2n+9 for all n \in \mathbb{N^{*}}.
    Quote Originally Posted by Sorombo View Post
    I found one, but only by trial and errors.

    f(n)=2n+3

    f(2n+3)=2(2n+3)+3=4n+9

    f(2n+3-n)=2(n+3)+3=2n+9
    If you assume the function is linear then it's easy. Let f(n) = ax + b and you'll get two equations in a and b. They're pretty easy to solve. The only possible result is 2n + 3. Now, why can't we do this with a quadratic?

    -Dan
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