Building a Piecewise Function with Elementary Functions

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Discussion Overview

The discussion revolves around constructing a piecewise function using elementary functions, specifically defining a function F: R -> R such that F(x) = x^2 for x ≤ a and F(x) = x^3 for x > a. Participants explore the implications of differentiability and the classification of such functions.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant questions whether the function can be expressed as a composition of elementary functions, suggesting that it cannot due to the lack of smoothness at the point a.
  • Another participant clarifies that the function is indeed piecewise defined and questions if such functions are considered elementary.
  • There is a suggestion that a Fourier Series could be an alternative approach, although its classification as elementary is debated.
  • A participant emphasizes that the function is not smooth at a, pointing out the differing third derivatives from the left and right at that point.
  • Multiple participants express confusion regarding the relevance of differentiability and the nature of piecewise functions.
  • There are assertions that while x^3 is infinitely differentiable, the constructed piecewise function is not, leading to further questioning of the need for a piecewise definition.

Areas of Agreement / Disagreement

Participants generally disagree on the necessity and classification of the piecewise function, with some advocating for its use while others question its appropriateness and explore alternatives. The discussion remains unresolved regarding the best approach to express the function.

Contextual Notes

There are unresolved questions about the definitions of elementary functions and the implications of differentiability at the transition point a. The discussion also touches on the smoothness of individual components versus the overall piecewise function.

roger
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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
 
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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.
 
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?
 
Or did you mean a piecewise function, with:

[tex]f(x) = \left\{ \begin{matrix}<br /> x^2, & \mbox{if } x \leq a\\<br /> x^3, & \mbox{if } x > a<br /> \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.
 
If you desperately don't want a piecewise function, you could try a Fourier Series, though I highly doubt that counts as elementary.
 
roger said:
I'm not sure I understand the relevance of your last comment about differentiability.

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).
 
but does anybody understand my question? someone mentioned Fourier series so is this the way forward to express it?
 
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.
 
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.
 
  • #10
Everybody resists piecewise at first, but eventually they all give in.
 
  • #11
roger said:
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.

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.
 
  • #12
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?
 
  • #13
roger said:
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.
"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!
 
  • #14
It's certainly the value I like best.
 

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