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Math Help - Find inverse and prove

  1. #1
    Junior Member
    Joined
    Jan 2007
    Posts
    26

    Question Find inverse and prove

    I'm having trouble with two problems that I'm stuck on, I think the fact that they involve fractions is confusing me...

    1) f(x) = (x+1)/x
    f^-1(y) = (x+1)/x
    yx = (x+1)
    yx-1 = x

    check:
    y = (yx-1)+1/(yx-1)

    and that's how far I got...

    2) f(x) = 2 - 1/(x+1)
    (y-2)(x+1) = -1
    x+1 = -1/(y-2)
    x = -1 (-1/y-2)

    check:
    y = 2 - 1/(-1(-1/y-2) - 1

    and stuck out of my mind here

    Any help would be great. Thanks
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  2. #2
    Member
    Joined
    Dec 2009
    Posts
    226
    To find the inverse function of y = f(x), switch the variables and solve for y.

    y = \frac{x + 1}{x}

    Switch the variables.

    x = \frac{y + 1}{y}

    Now, solve for y. First, separate the fraction.

    x = \frac{y}{y} + {1}{y}

    Reduce \frac{y}{y}.

    x = 1 + \frac{1}{y}

    Subtract 1 from the equation.

    x - 1 = \frac{1}{y}

    Take the reciprocal of the equation.

    \frac{1}{x - 1} = y

    Therefore, the inverse function of f(x) is y = \frac{1}{x - 1}.

    Use the same procedure for the second function.

    y = 2 - \frac{1}{x + 1}

    Switch the variables.

    x = 2 - \frac{1}{y + 1}

    Subtract 2 from the equation.

    x - 2 = -\frac{1}{y + 1}

    Reverse the sign of the equation.

    2 - x = \frac{1}{y + 1}

    Take the reciprocal of the equation.

    \frac{1}{2 - x} = y + 1

    Subtract 1 from the equation.

    \frac{1}{2 - x} - 1 = y
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