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Math Help - Inverse of a function with natural log

  1. #1
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    Inverse of a function with natural log

    I am asked to find the inverse of f(x)=ln(x+\sqrt{x^2 +1})

    I understand the basics of how to find the inverse but I get stuck on solving this one. Here is what I have so far.

    y=ln(x+\sqrt{x^2 +1})
    e^y =x+ \sqrt{x^2+1}

    I'm not sure how to proceed from here. I thought maybe squaring both sides would be the next step but I can never come up with the correct answer. How do I solve this?
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  2. #2
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    Re: Inverse of a function with natural log

    Quote Originally Posted by BobRoss View Post
    I am asked to find the inverse of f(x)=ln(x+\sqrt{x^2 +1})

    I understand the basics of how to find the inverse but I get stuck on solving this one. Here is what I have so far.

    y=ln(x+\sqrt{x^2 +1})
    e^y =x+ \sqrt{x^2+1}

    I'm not sure how to proceed from here. I thought maybe squaring both sides would be the next step but I can never come up with the correct answer. How do I solve this?
    \displaystyle \begin{align*} e^y &= x + \sqrt{x^2 + 1} \\ e^y - x &= \sqrt{x^2 + 1} \\ \left(e^y - x \right)^2 &= x^2 + 1 \\ e^{2y} - 2x\,e^y + x^2 &= x^2 + 1 \\ e^{2y} - 2x\,e^y &= 1 \\ e^{2y} - 1 &= 2x\,e^y \\ e^y - e^{-y} &= 2x \\ \frac{1}{2}\left( e^y - e^{-y} \right) &= x \end{align*}
    Thanks from BobRoss
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    Re: Inverse of a function with natural log

    What is happening when you go from:

    e^{2y} -1=2xe^y

    to

    e^y - e^{-y} =2x

    ?
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    Re: Inverse of a function with natural log

    Quote Originally Posted by BobRoss View Post
    What is happening when you go from:

    e^{2y} -1=2xe^y

    to

    e^y - e^{-y} =2x

    ?
    Dividing both sides by \displaystyle \begin{align*} e^y \end{align*}.
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    Re: Inverse of a function with natural log

    Ahh right that's what I thought but I just got confused for a second. Thanks, this helps a ton!
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    Re: Inverse of a function with natural log

    I have another inverse function question that I am having trouble with.

    f(x)=arctan(\frac{x-1}{x+1})

    I don't even really know how to start solving this one.
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    Re: Inverse of a function with natural log

    Quote Originally Posted by BobRoss View Post
    I have another inverse function question that I am having trouble with.

    f(x)=arctan(\frac{x-1}{x+1})

    I don't even really know how to start solving this one.
    x = \arctan\left(\frac{y-1}{y+1}\right)

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

    solve for y



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