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Math Help - question regarding the composition of functions

  1. #1
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    question regarding the composition of functions

    Simple question. Suppose f(g(x)) = g(f(x)). Then must one of the following be true?

    1. One of the functions is the identity.

    2. The two functions are inverses.
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  2. #2
    Grand Panjandrum
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    Re: question regarding the composition of functions

    Quote Originally Posted by icemanfan View Post
    Simple question. Suppose f(g(x)) = g(f(x)). Then must one of the following be true?

    1. One of the functions is the identity.

    2. The two functions are inverses.
    Consider the special case f(x) and g(x) are the same function.

    CB
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  3. #3
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    Re: question regarding the composition of functions

    Suppose then the criteria become:

    1. One of the functions is the identity

    2. The two functions are inverses

    3. The two functions are the same

    Must one of these statements be true?
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  4. #4
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    Re: question regarding the composition of functions

    What if f(x)= ax, g(x)= bx, where a and b are constants?
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  5. #5
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    Re: question regarding the composition of functions

    f o g:x = g o f:x if one of the following statement holds


    i) one of the functions is the identity
    ii)the two function are inverses
    iii) the two functions are the same
    iv) the two functions are of the form f(x) = ax + b = x + b that is a =1
    g(x) = cx + d = x + d that is b = 1

    we call this functions translations

    v) f and g have the same fixed point.
    fixed point is the point X0 = (xo,xo) such that f(xo) = xo and g(x0) = xo


    fixed point of f = -b/(a-1)
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