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Very confusing question on Compositional Functions and natural numbers Watch

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    Hi,
    I've been trying to figure out this question for a few days now, to no avail.

    The question:

    the function f(x) = 2x

    g(f(x) = x

    Find a function for g() that satisfies the above equation and that when a natural number is input into the g function, it produces only a natural number.

    I've been stuck on this for days, and the only help I could get from my lecturer was "the question explains itself". Any guidance or pointers?
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    (Original post by thomoski2)
    I've been stuck on this for days, and the only help I could get from my lecturer was "the question explains itself". Any guidance or pointers?
    I would be very interested in seeing the opinons of those wiser and more experiened than me, because as far as I can see, this doesn't make any sense at all! Unless there is some extra information about the domain of f that we should know about.
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    (Original post by Pangol)
    I would be very interested in seeing the opinons of those wiser and more experiened than me, because as far as I can see, this doesn't make any sense at all! Unless there is some extra information about the domain of f that we should know about.
    It might have been my explanation of the question, just to clarify, he's an image of the original question (underlined in red)
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    (Original post by thomoski2)
    x
    You can't use g(x)=\dfrac{x}{2} since this wouldn't map odd x to natural numbers. But if we can use floor and ceiling functions I think you could come up with a way around that.
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    (Original post by thomoski2)
    It might have been my explanation of the question, just to clarify, he's an image of the original question (underlined in red)
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    so then does g(2x)=x

    so in other words g(x)= x/2 ??
    then again it's not natural if you got fractions -__-
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    (Original post by will'o'wisp2)
    so then does g(2x)=x

    so in other words g(x)= x/2 ??
    then again it's not natural if you got fractions -__-
    That's what I initially thought, and tried, but hit the same wall
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    (Original post by I hate maths)
    You can't use g(x)=\dfrac{x}{2} since this wouldn't map odd x to natural numbers. But if we can use floor and ceiling functions I think you could come up with a way around that.
    I did think about this, as I've used them in programming before, but I'm not 100% sure how to go about using them in theoretical maths
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    (Original post by thomoski2)
    I did think about this, as I've used them in programming before, but I'm not 100% sure how to go about using them in theoretical maths
    In mathematics and computer science, the floor function is the function that takes as input a real number and gives as output the greatest integer that is less than or equal to . Similarly, the ceiling function maps to the least integer that is greater than or equal to . [Cited from Wikipedia]. It's literally the same but the application I'm also uncertain of.
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    (Original post by AmmarTa)
    In mathematics and computer science, the floor function is the function that takes as input a real number and gives as output the greatest integer that is less than or equal to . Similarly, the ceiling function maps to the least integer that is greater than or equal to . [Cited from Wikipedia]. It's literally the same but the application I'm also uncertain of.
    Yeah, sounds the same as I thought, I might have a look into using them then.
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    (Original post by thomoski2)
    ...
    There's no requirement to have a single simple formula for g(x).

    You could define g(x) = x/2, for x even, and anything you like for x odd (as long as it's a natural number).
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    (Original post by ghostwalker)
    There's no requirement to have a single simple formula for g(x).

    You could define g(x) = x/2, for x even, and anything you like for x odd (as long as it's a natural number).
    Yeah, I initially tried a simple formula, but the only thing I could come up with was x/2, which obviously wouldnt work. The trouble I had was, without a proper way to use floor and ceiling methods, I wouldn't even know where to start on what I presume would be a very long, complicated formula to essentially divide without dividing
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    (Original post by thomoski2)
    Yeah, I initially tried a simple formula, but the only thing I could come up with was x/2, which obviously wouldnt work. The trouble I had was, without a proper way to use floor and ceiling methods, I wouldn't even know where to start on what I presume would be a very long, complicated formula to essentially divide without dividing
    It wouldn't be complicated, in fact quite simple.

    You're only interested in what happens to even values, and x/2 will return an integer, and be unaffected by the floor function, for example.

    For odd values the floor function would convert x/2 to an integer - doesn't matter what it is.
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    (Original post by thomoski2)
    It might have been my explanation of the question, just to clarify, he's an image of the original question (underlined in red)
    Yes, this does make quite a bit of difference, since it is now clear that the domain of f is N, not R as I had assumed. The ideas discussed above are what you need!
 
 
 
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