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Thread: Mathematical Sum of Functions Problem

  1. #1 Mathematical Sum of Functions Problem 
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    I was recently posed the problem of expressing (x - 1)/(x + 1) as the sum of an even and odd function. The person who posed the problem suggested that I couldn't solve it.

    This is how I went about doing the problem:

    First, let's model our problem:











    Furthermore, let us write a system of equations:





    Now we can attempt to solve for g(x):























































    Can someone check my solution please? This was a problem that took me a decent amount of time and I would appreciate it. Also, does anyone know where this problem came from? As it was one of the more difficult ones I've encountered, I doubt that the person that posed it developed the problem either.


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  3. #2 Re: Mathematical Sum of Functions Problem 
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    Quote Originally Posted by Ellatha
    I was recently posed the problem of expressing (x - 1)/(x + 1) as the sum of an even and odd function. The person who posed the problem suggested that I couldn't solve it.

    This is how I went about doing the problem:

    First, let's model our problem:











    Furthermore, let us write a system of equations:





    Now we can attempt to solve for g(x):























































    Can someone check my solution please? This was a problem that took me a decent amount of time and I would appreciate it. Also, does anyone know where this problem came from? As it was one of the more difficult ones I've encountered, I doubt that the person that posed it developed the problem either.
    I am not about to read that gigantic post, but ANY function is the sum of an even and an odd function:

    F(x) = 1/2[ F(x) + F(-x)] + 1/2[F(x) - F(-x)]


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  4. #3  
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    Is my solution correct though?

    I said that f(x) + g(x) = h(x)







    By the way the question was to express the given function as the sum of an even and odd function (and not necessarily prove any function is the sum of an even and odd function).

    EDIT: syntax fixed.
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  5. #4  
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    Quote Originally Posted by Ellatha
    Is my solution correct though?

    I said that f(x) + g(x) = h(x)







    By the way the question was to express the given function as the sum of an even and odd function (and not necessarily prove any function is the sum of an even and odd function).
    You seem a bit lazy. DrRocket's answer is correct and trivial to implement. Why not try it and see if you get the same answer. If not, you made a mistake someplace.
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  6. #5  
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    Quote Originally Posted by mathman
    You seem a bit lazy. DrRocket's answer is correct and trivial to implement. Why not try it and see if you get the same answer. If not, you made a mistake someplace.
    0_o.

    Who the hell do you think you are commenting in my thread and calling me lazy? Do not make trivial remarks about my personal character; you don't know me, don't ever pretend that you do, and don't ever forget that.
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  7. #6  
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    Quote Originally Posted by Ellatha
    Is my solution correct though?

    I said that f(x) + g(x) = h(x)







    By the way the question was to express the given function as the sum of an even and odd function (and not necessarily prove any function is the sum of an even and odd function).
    But you should be able to see that the general solution is quite easy -- a lot easier than whatever it is that you did (I have no intention of working through it, it is unnecessarily tedious though it may be right).

    Why not just insert your specific function into the general expression and see if it agrees with your final answer ?
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  8. #7  
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by Ellatha
    Is my solution correct though?

    I said that f(x) + g(x) = h(x)







    By the way the question was to express the given function as the sum of an even and odd function (and not necessarily prove any function is the sum of an even and odd function).
    But you should be able to see that the general solution is quite easy -- a lot easier than whatever it is that you did (I have no intention of working through it, it is unnecessarily tedious though it may be right).

    Why not just insert your specific function into the general expression and see if it agrees with your final answer ?
    Okay, I will.
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  9. #8  
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    Quote Originally Posted by Ellatha
    for one,

    and two, the final answer would actually be f(x)+g(x)=1\ne h(x) so you are not correct.
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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  10. #9  
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    Quote Originally Posted by Arcane_Mathematician
    Quote Originally Posted by Ellatha
    for one,

    and two, the final answer would actually be f(x)+g(x)=1\ne h(x) so you are not correct.
    I meant (as you can see by looking at the OP)

    The question was to express as the sum of an even and odd function. In this case, the odd function is g(x), that is , and f(x) is the even function, .
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  11. #10  
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    yes, that works nicely, and you have a solution that works
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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