Functional equations are weird

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This topic contains 2 replies, has 2 voices, and was last updated by  Agamdeep Singh 2 days, 20 hours ago.

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  • #29158

    Agamdeep Singh
    Participant

    Is the following method correct?

    I mean is it enough to imply that f(n)=N

    Do we have to prove wether f is ever increasing or ever decreasing before doing this. Like could it be that f is sometimes greater than n and sometimes less, and we proved that is not greater when it is less.

    Kind of like sin(x). It increases and decreases. Do we have to consider the that f might also be increasing and decreasing above and below n.

    or is it correct and that is taken care of when we take the case f(n)>n.

    Help would really be appreciated.

    Thanks.

    #29183

    Nitin Prasad
    Participant

    No, just checking for the case where f is decreasing or increasing is not sufficient. As you pointed out, we need to consider the case (if possible) where f is just some random arrangement and doesn’t have monotonic property.

    Hints

    • If ‘f’ a one-one function? Observe that if f(m)=f(n) implies m=n then, the given function is one-one.
    • Observe that $$f[ f(m+k)+f(n-k) ]=m+n=f[ f(m)+f(n) ]$$. What can we conclude?

    Using further manipulations you can finally show that f(n)=n. The complete solution has been attached below

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    #29266

    Agamdeep Singh
    Participant

    Thanks sir

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