# Constructing a function

1. Jul 10, 2007

### roger

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. Jul 10, 2007

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.

3. Jul 10, 2007

### roger

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?

4. Jul 10, 2007

### Parthalan

Or did you mean a piecewise function, with:

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

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

5. Jul 11, 2007

### Gib Z

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

6. Jul 11, 2007

### matt grime

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).

7. Jul 11, 2007

### roger

but does anybody understand my question? someone mentioned fourier series so is this the way forward to express it?

8. Jul 11, 2007

### roger

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.

9. Jul 11, 2007

### Gib Z

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. Jul 11, 2007

### ObsessiveMathsFreak

Everybody resists piecewise at first, but eventually they all give in.

11. Jul 11, 2007

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. Jul 11, 2007

### matt grime

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

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. Jul 11, 2007

### HallsofIvy

Staff Emeritus
"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

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"0" is a perfectly good value!

14. Jul 19, 2007

### disregardthat

It's certainly the value I like best.