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Help with inverse function problem

  1. Jan 13, 2009 #1
    1. The problem statement, all variables and given/known data

    If f(x) = x + cos(x), find f-inverse of (1)

    2. Relevant equations

    **first time asking a forum question, please inform me of any errors in posting this question

    3. The attempt at a solution

    (1) y = x + cos(x) => if y = f(x)

    (2) x = y + cos(y) => if x = f(y)

    (3) ** solve for y? this is where I am stuck, is it possible to completely show work and solve for "y" before plugging in "1"? I tried switching the variables and using the solve command on my ti-89 and it gives the answer:
    cos(y) + y = x
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jan 13, 2009 #2

    jgens

    User Avatar
    Gold Member

    I don't think there is anyway of expressing the inverse of that function in terms of elementary function (someone with a more authoritative math background could clarify). However, if we note the definition of an inverse function: f:X -> Y than f^-1:Y -> X we can find a solution for f^-1(1). Hence, to find f^-1(1) all we need find is the x value which produces an f(x) value of 1; therefore, 1 = x + cos(x). Can you solve that?
     
  4. Jan 13, 2009 #3
    Ok gotcha, x = 0, thanks for the help
     
  5. Jan 13, 2009 #4

    jgens

    User Avatar
    Gold Member

    Yep, so f^-1(1) = 0.
     
  6. Jan 13, 2009 #5
    There is no other solution to f(x) = 1 because f is nondecreasing everywhere and increasing at 0 (use f' to show this).
     
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