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Homework Help: Inverse of a function

  1. Feb 7, 2010 #1
    1. The problem statement, all variables and given/known data

    f(-1,+1) -> R is given by f(x) = x/(1 - |x|)

    Find the inverse of f.

    Are f and the inverse of f continuous?

    2. Relevant equations

    3. The attempt at a solution

    I have shown that f is 1-1.
    f((-1,+1)) -> (-[tex]\infty[/tex], +[tex]\infty[/tex])

    Let y = f-1(x), so f(y) = x

    y/ (1- |y|) = x
    y = x(1- |y|)

    If y = 0 x = 0
    If y> 0 y =x(1-y), so y = x/(1+x)
    If y< 0 y =x(1+y), so y = x/(1-x)

    Can I say if y<0 then x <0 and if y >0 then x >0?
    Then y=x/(1+|x|)

    Would the domain of the inverse be (-[tex]\infty[/tex],+[tex]\infty[/tex])?

    If this is so, f and the inverse of f are continuous as I know f(x) =x, f(x) = 1, f(x) = |x| are continuous and if f and g are continuous then so are |f|, f+g, f-g, and f/g.
  2. jcsd
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