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Math Help - Find the Inverse Function

  1. #1
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    Thumbs up Find the Inverse Function

    Hello,

    I took this a long time ago and I need to refresh my mind about it, can you please help ?

    The function y=f(x) is defined by y=x^2 + 4 for  -2< x < 0.
    Find the inverse function f^-1(x) stating its range and domain, and sketch the inverse function f^-1(x).

    Now I do now really want the sketch since I will try finding that out myself.
    But how do we find f^-1(x) ?? What I remember is that im supposed to re-arrange the equation so that it is x= .. but wont that change the domain and range ?

    Please help,

    Kind Regards,

    Z
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  2. #2
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    e^(i*pi)'s Avatar
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    The range and the domain swap for the inverse function.

    Thus if you find the range of the original function it will be the domain of the inverse.
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  3. #3
    MHF Contributor Calculus26's Avatar
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    Domain of f^-1 is the range of f

    Range of f^-1 is the domain of f




    is the range of f^-1 this will be important as

    x = + sqrt(y-4)

    x= - sqrt(y-4)

    f^-1(x) = -sqrt( x-4) Domain 4 < x < 8

    Note the range is then -2 < y < 0
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  4. #4
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    Thanks for the reply guys, thank you Calculus26 for the explanation.

    Can you please tell me why you chose the negative sign at x= - sqrt(y-4) ??

    Thanks
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  5. #5
    MHF Contributor Calculus26's Avatar
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    is the range of f^-1 this will be important as

    x = + sqrt(y-4)
    the range is negative
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  6. #6
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    Oh so the graph will be a negative .. of course ..

    I will solve the question now and post the sketch

    Thanks
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  7. #7
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    Ok I managed to get it all except for the range, why is y<8 ??
    It is greater than 4 because Square root of 0 is undefined, so it must be greater than 4 ..

    But any number greater than 8 will still give an answer ?
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  8. #8
    MHF Contributor Calculus26's Avatar
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    Consider the original problem

    The domain was restricted to -2 < x < 0

    Which gave a restricted range 4 < y < 8

    These restrictions carry over to the inverse with the domains and ranges
    reversed.
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  9. #9
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    Oh yeahhhh

    How can I forget such thing, thank you very very much !!
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  10. #10
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    e^(i*pi)'s Avatar
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    As a check the inverse of a function will be it's reflection in the line y=x
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  11. #11
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    Yes I realized that when I sketched the function
    Ill add that as a note for future reference, thank you.
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