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Prove this inequality

  1. May 10, 2013 #1

    utkarshakash

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    1. The problem statement, all variables and given/known data
    If f and g are two distinct linear functions defined on R such that they map[-1,1] onto [0,2] and h:R-{-1,0,1}→R defined by h(x)=f(x)/g(x) then show that |h(h(x))+h(h(1/x))|>2

    2. Relevant equations

    3. The attempt at a solution
    I assume f(x) to be ax+b and g(x) to be lx+m so that h(x) is (ax+b)/(lx+m). From here I can write h(h(x)) and h(h(1/x)) but there is nothing I can see that will help me to prove this inequality. Any ideas?
     
  2. jcsd
  3. May 11, 2013 #2

    SammyS

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    You can be more specific regarding the functions, f & g .

    There are only two distinct linear functions which map [-1,1] onto [0,2] .

    What are they?
     
  4. May 11, 2013 #3

    utkarshakash

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    y=x+1 and y=-x+1. Are these correct?
     
  5. May 11, 2013 #4

    SammyS

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    Yes. Equivalently, y=1+x and y=1-x .

    So, there are only two cases to consider.

    Choosing f(x) = 1+x and g(x) = 1-x, for now, what is h(1/x) ?
     
  6. May 11, 2013 #5

    utkarshakash

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    x+1/x-1 which can be reduced to -f(x)/g(x)

    And after simplifying further I am left with proving this inequality

    |x-(1/x)|>2
     
    Last edited: May 11, 2013
  7. May 12, 2013 #6

    SammyS

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    It looks like you get h(1/x) = -f(x)/g(x), for either way of assigning 1-x and 1+x to f(x) and g(x).
    I got something similar to x-(1/x), but it is different.

    Check your algebra.

    It seems to me that one could do this more abstractly using some calculus, etc. (Maybe the Calculus part comes in the next step)
     
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