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Math Help - Inverse of this polynomial function?

  1. #1
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    Inverse of this polynomial function?

    f(x) = sqrt(x3 - 21x2 + 147x)

    Show all of your work and explain how you did every step. Include any equations that you used.
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  2. #2
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    If I remember correctly, for a function to have an inverse it has to pass the horizontal line test (A horizontal line drawn on the graph has to correspond to a single value), if you plot this function you will find that it does not pass the horizontal line test. So it cannot have an inverse function.
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  3. #3
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    Hello ericforman65

    Welcome to Math Help Forum!
    Quote Originally Posted by ericforman65 View Post
    f(x) = sqrt(x3 - 21x2 + 147x)

    Show all of your work and explain how you did every step. Include any equations that you used.
    Let y = \sqrt{x^3 - 21x^2 + 147x}

    Then:
    y^2 = x^3 - 21x^2 + 147x
    If we are to stand any chance of finding the inverse, we must assume that we can find an integer a, such that:
    (x-a)^3 = x^3 - 21x^2 + 147x -a^3
    Expanding the LHS and comparing coefficients gives a = 7. So:
    y^2 -7^3=(x-7)^3
    \Rightarrow x = (y^2-343)^{\frac13}+7
    So the inverse function is:
    f^{-1}(x)=(x^2-343)^{\frac13}+7
    Grandad
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  4. #4
    Member mathemagister's Avatar
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    Quote Originally Posted by bilalsaeedkhan View Post
    If I remember correctly, for a function to have an inverse it has to pass the horizontal line test (A horizontal line drawn on the graph has to correspond to a single value), if you plot this function you will find that it does not pass the horizontal line test. So it cannot have an inverse function.
    No, in this case, because of the square root, the function does pass the horizontal line test.
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  5. #5
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    Thanks but I have a question, isn't the graph of an inverse function a reflection across the line y=x?
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  6. #6
    Member mathemagister's Avatar
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    Quote Originally Posted by bilalsaeedkhan View Post
    Thanks but I have a question, isn't the graph of an inverse function a reflection across the line y=x?
    Yes, it is, but unless the original function fails the horizontal line test, the inverse will pass the vertical line test and be a function.
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  7. #7
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    Quote Originally Posted by Grandad View Post
    Hello ericforman65

    Welcome to Math Help Forum!Let y = \sqrt{x^3 - 21x^2 + 147x}

    Then:
    y^2 = x^3 - 21x^2 + 147x
    If we are to stand any chance of finding the inverse, we must assume that we can find an integer a, such that:
    (x-a)^3 = x^3 - 21x^2 + 147x -a^3
    Expanding the LHS and comparing coefficients gives a = 7. So:
    y^2 -7^3=(x-7)^3
    \Rightarrow x = (y^2-343)^{\frac13}+7
    So the inverse function is:
    f^{-1}(x)=(x^2-343)^{\frac13}+7
    Grandad
    Thanks so much, but you lost me a little in there. I have to be able to explain every single thing that I did and why I did it in a written response. Could you please explain very much in detail everything that you did after:
    y^2 = x^3 - 21x^2 + 147x
    Thanks for any future response.

    PS - The work is perfect and I also don't know what LHS is.
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  8. #8
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by ericforman65 View Post
    Thanks so much, but you lost me a little in there. I have to be able to explain every single thing that I did and why I did it in a written response. Could you please explain very much in detail everything that you did after:
    y^2 = x^3 - 21x^2 + 147x
    Thanks for any future response.

    PS - The work is perfect and I also don't know what LHS is.
    Grandad has given ( he always gives) the best possible explanation. I will try to explain the steps. Lets see if I succeed.

    y^2 = x^3 - 21x^2 + 147x.................(1)

    LHS= Left hand side of the equation. In this case, y^2

    RHS = Right hand side of the equation. In this case, x^3 - 21x^2 + 147x

    Do you know that
    (a-b)^{3} = a^{3} - 3.a^{2}.b + 3.a.b^{2}- b^{3}......(2)

    Try to transform RHS into this form.

    RHS = x^{3} - 3.(x^{2})(7) + 3(x)(49)

    If you compare (2) with RHS above, you will see that you can transform RHS

    into the form of (a-b)^3, where a = x, and b = 7. However,

    you have to notice that you do not have -(7^3), which is equal

    to 343 on the RHS. So you will have to subtract 343 on both sides of equation(1).

    so,

    y^2 - 343 = x^{3} - 3.(x^{2})(7) + 3(x)(49)-343

    Therefore,

    y^2 - 7^3 = (x-7)^3

    (x-7) = (y^2 - 7^3)^{\frac{1}{3}}

    x = (y^2 - 7^3)^{\frac{1}{3}} + 7

    which is the inverse of the function
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