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Math Help - Inverse of x+sinx

  1. #1
    Senior Member pankaj's Avatar
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    Inverse of x+sinx

    Determine the inverse of the function:
     <br />
f(x)=x+sinx<br />
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by pankaj View Post
    Determine the inverse of the function:
     <br />
f(x)=x+sinx<br />
    First you must restrict your domain so that your function is one to one, and I am not sure this can be found? What makes you think it can? I mean we know that this has an inverse, but what makes you think it can be expressed in elementary terms?
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  3. #3
    Senior Member pankaj's Avatar
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    The function is both one-one as well as onto since
    f'(x)=1+cosx\geq 0 for all real x.i.e. f(x) is an incresing function and will never repeat its value
    From the graph it appears as if the inverse is f^{-1}(x)=x-sinx since it appears to be reflection of f(x) with respect to the line y=x as mirror.I am not sure about this though.
    I am looking for verification
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by pankaj View Post
    The function is both one-one as well as onto since
    f'(x)=1+cosx\geq 0 for all real x.i.e. f(x) is an incresing function and will never repeat its value
    From the graph it appears as if the inverse is f^{-1}(x)=x-sinx since it appears to be reflection of f(x) with respect to the line y=x as mirror.I am not sure about this though.
    I am looking for verification
    You are right, I was not thinkin clearly. But I do not think that f^{-1}(x)=x-\sin(x), because for example f(f^{-1}(2))\ne{2}
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