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Math Help - Please Help - statements about functions

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    Please Help - statements about functions

    In front of you four statements that regard to functions f:R-->R.

    1. Exist a>0 and exist x in R so that f(x+a)=f(x)

    2. For every a>0, exist x in R so that f(x+a)=f(x)

    3. Exist a>0 so that for every x in R so thatf(x+a)=f(x)

    4. For every a>0 and every x in R so that f(x+a)=f(x)

    Questions:

    i. Characterize all functions from 1.
    ii. Characterize all functions from 4.
    iii. Find a function that is 1, but not 2,3,4
    iv. Find a function that is 2, but not 3 and 4.
    v. Find a function that is 3, but not 2 and 4.




    Thanks!
    Last edited by mr fantastic; January 3rd 2010 at 11:07 PM. Reason: Changed post title
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  2. #2
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    Quote Originally Posted by Also sprach Zarathustra View Post
    In front of you four statements that regard to functions f:R-->R.

    1. Exist a>0 and exist x in R so that f(x+a)=f(x)

    2. For every a>0, exist x in R so that f(x+a)=f(x)

    3. Exist a>0 so that for every x in R so thatf(x+a)=f(x)

    4. For every a>0 and every x in R so that f(x+a)=f(x)

    Questions:

    i. Characterize all functions from 1.
    I take it you mean all functions that have property #1. Since a> 0 that simply means that there exist two distinct numbers, x and y, such that f(x)= f(y).

    ii. Characterize all functions from 4.
    Let x and y be any two numbers. Then we can take a= |y-x| so that f(x)= f(x+a)= f(y).

    iii. Find a function that is 1, but not 2,3,4
    There are many many easy examples. Try!

    iv. Find a function that is 2, but not 3 and 4.
    v. Find a function that is 3, but not 2 and 4.




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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    I am not understanding this! What I need to show...? Can you please give more detailed answer?
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  4. #4
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    For (i) you simply have functions that are not "one to one".

    For (i), as I said, you must have f(x)= f(y) for all x and y- a constant function.
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  5. #5
    MHF Contributor Also sprach Zarathustra's Avatar
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    Yes, this (i) is easy, but what about (ii), (iii), (iv) and (v)?

    Thank you for your time and knowledge!
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