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Math Help - Functions

  1. #1
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    Functions

    Hello! I don't really understand how to answer the second part of the question (the "hence" part). Please help! Thank you!

    f(x) = 2 - (1/4)x^3 , x is a real number, x>0

    Sketch the graph of f and find its range. Hence state the set of values of x for which ff^-1(x) = f^-1f(x)

    [By f^-1(x) I mean f inverse. I am not sure of how to type out f inverse on the computer... ]

    Thank You!
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  2. #2
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    Quote Originally Posted by Tangera View Post
    Hello! I don't really understand how to answer the second part of the question (the "hence" part). Please help! Thank you!

    f(x) = 2 - (1/4)x^3 , x is a real number, x>0

    Sketch the graph of f and find its range. Hence state the set of values of x for which ff^-1(x) = f^-1f(x)

    [By f^-1(x) I mean f inverse. I am not sure of how to type out f inverse on the computer... ]

    Thank You!
    Provided you have a 1-to-1 function (which you do in this case), f(f^{-1}(x)) = x\, over \text{dom}f^{-1} and \, f^{-1}(f(x)) = x\, over \text{dom} f.

    So for f(f^{-1}(x)) = f^{-1}(f(x)) you need the set of values of x such that \text{dom}f = \text{dom}f^{-1}.

    But you know that for any 1-to-1 function f, \text{dom}f^{-1} = \text{ran} f.

    So you need the set of values of x such that \text{dom}f = \text{ran}f. From the graph you can see that these values will be 0 \leq x \leq 2.

    So f(f^{-1}(x)) = f^{-1}(f(x)) = x for 0 \leq x \leq 2.
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  3. #3
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    erm...may I know what is dom f and ran f? Thank you!
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    Quote Originally Posted by Tangera View Post
    erm...may I know what is dom f and ran f? Thank you!
    It is shorthand:

    dom f is the domain of the function f.

    ran f is the range of the function f.
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  5. #5
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    Ahhh...I see. Thank you!
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