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Constructing a function

  1. Jul 10, 2007 #1
    How would one construct a function involving elementary functions, F:R->R such that F(x)=x^2 iff x<=a and F(x)=x^3 iff x>a?


    cheers,
    roger
     
  2. jcsd
  3. Jul 10, 2007 #2
    i'm not sure what you are trying to say. What you have given is a function expressed in terms of elementary functions.

    Do you mean to ask if it can be expressed as a composition of elementary functions? I would think not, as such a composition would be inifnitely differentiable, where as this function is not.
     
  4. Jul 10, 2007 #3
    yes I guess I mean as a composition of elementary functions, however I'm not sure I understand the relevance of your last comment about differentiability.

    can anyone help me?
     
  5. Jul 10, 2007 #4
    Or did you mean a piecewise function, with:

    [tex]f(x) = \left\{ \begin{matrix}
    x^2, & \mbox{if } x \leq a\\
    x^3, & \mbox{if } x > a
    \end{matrix}[/tex]

    I don't know if functions defined like this are considered elementary, but in this case, [itex]f:\mathbb{R} \rightarrow \mathbb{R}[/itex]. I'm not sure from your original post if you are talking about the two functions, [itex]x^2[/itex] and [itex]x^3[/itex] being elementary, or the new function.
     
  6. Jul 11, 2007 #5

    Gib Z

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    If you desperately don't want a piecewise function, you could try a Fourier Series, though I highly doubt that counts as elementary.
     
  7. Jul 11, 2007 #6

    matt grime

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    The elementary functions, and therefore any composition of them, are smooth, was his point. Your function is not smooth at a. Trying to work out the third derivatives at a from the left and right gives 0 and 6. These are not equal, the function is _at most_ twice differentiable, even assuming that you correct it so that it is continuous (a must be 0 or 1).
     
  8. Jul 11, 2007 #7
    but does anybody understand my question? someone mentioned fourier series so is this the way forward to express it?
     
  9. Jul 11, 2007 #8
    Matt I read somewhere something along the lines that a function like x^3 can be differentiated as many times as one wishes is this correct? even once you get to 6.
     
  10. Jul 11, 2007 #9

    Gib Z

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    That is not elementary...and quite unnecessary I would think..What is wrong with piecewise?

    EDIT: In response to your last post, yes your pieces individually are infinitely differentiable, but put together they are not.
     
  11. Jul 11, 2007 #10
    Everybody resists piecewise at first, but eventually they all give in.
     
  12. Jul 11, 2007 #11
    If you are not 100 percent certain of your answer to this question, you are probably to early on to worry about elementary vs. transcendental functions.
     
  13. Jul 11, 2007 #12

    matt grime

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    1. x^3 from R to R certainly is smooth.

    2. Your function isn't x^3.

    I echo Deadwolfe here - if you can't see these points, then why are you attempting to get someone to teach you fourier analysis?
     
  14. Jul 11, 2007 #13

    HallsofIvy

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    "even once you get to 6" what?

    f(x)= x3

    f '(x)= 3x2

    f '''(x)= 6x

    f ''''(x)= 6

    f '''''(x)= 0

    f ''''''(x)= 0

    f ''''''(x)= 0

    .
    .
    .


    "0" is a perfectly good value!
     
  15. Jul 19, 2007 #14

    disregardthat

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    It's certainly the value I like best.
     
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