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Inverse function help

  1. May 6, 2008 #1
    Ok, I decided to review basic algebra since I haven't done anything with it in like, forever. I came across an inverse function problem that I can't get the right answer.

    the equation is:

    y = cuberoot(x+sqrt(1+x^2)) + cuberoot(x-sqrt(1+x^2))

    I tried replacing X with Y, and solving for Y
    and getting rid of the cube roots by cubing both sides
    X^3 = y + sqrt(1+y^2) + y - sqrt(1+y^2)
    simplifying a bit (the square roots go away)
    x^3 = 2y
    y = (1/2)x^3

    Yet the book I'm using says the answer is y=(1/2)(3x+x^3)

    What did I do wrong?
  2. jcsd
  3. May 6, 2008 #2


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    Staff Emeritus
    Science Advisor

    (a+ b)3 is NOT a3+ b3

    It is a3+ 3a2b+ 3ab2+ b3
  4. May 6, 2008 #3


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    Gold Member

    Your equation is a tad confusing, you have y = [tex]\sqrt[3]{x+sqrt(1+x^2)}[/tex] + [tex]\sqrt[3]{x-sqrt(1+x^2)}

    then you cube both sides and switch y's and x's to get [tex]x^{3}[/tex] = y+[tex]\sqrt(1+y^2)[/tex] + y-[tex]\sqrt(1+y^2)[/tex]

    Which is wrong. Now if your equation was y = [tex]\sqrt[3]{x+\sqrt(1+x^2)+x-\sqrt(1+x^2)}[/tex] this method would be correct. However, it would have been simplified easily before you even cube both sides; In this case it would be in the form y = [tex]\sqrt[3]{2x}[/tex] for the roots cancel automatically.
    Last edited: May 6, 2008
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