5
$\begingroup$

Is there a short way to say $f(f(f(f(x))))$?

I know you can use recursion:

$g(x,y)=\begin{cases} f(g(x,y-1)) & \text{if } y > 0, \ \newline x & \text{if } y = 0. \end{cases}$

  • 3
    I write $f^4(x)$ but you can write anything you want.2010-10-27

4 Answers 4

10

I personally prefer $f^{\circ n} = f \circ f^{\circ n-1} = \dotsb = \kern{-2em}\underbrace{f \circ \dotsb \circ f}_{n-1\text{ function compositions}}$

  • 2
    I should probably remark that the LaTeX above contains a dirty dirty hack: the underbrace construct is manually moved to the left with `\kern{-2em}`. One can only hope that MathJax picks up on the mathtools package and its `\mathclap` command.2010-10-27
  • 0
    why "n-1 function compositions" instead of n?2010-10-28
  • 0
    I like this one. It seems quite clear. But Sparr is right, should be n2010-10-28
  • 4
    Because if you want $f^{\circ 2}$, you're composing _once_: $f \circ f$. So the $n-1$ is counting compositions, not how many times it says $f$.2010-10-28
2

Some will say $f^4(x)$. But it becomes confused with the fourth power or fourth derivative of $f(x)$. I'm not sure what you mean here by "piecewise". To me "piecewise" would be something like a step function:

$f(x)=1$ if $x\gt 0$

$f(x)=0$ if $x\le 0$

  • 0
    Ross, that's what Roman numerals are for! $f^{\mathrm{IV}}$ for differentiation.2010-10-27
  • 1
    @muad Are you being serious?2010-10-27
  • 0
    By "piecewise", I meant g(x,y) { with the second line on top, and the third line on bottom. I think I've figured out how to fix it...2010-10-27
  • 1
    @muntoo, it's called Lagrange notation.2010-10-27
  • 4
    @muad Or you could use $f^{(x)}$ for differentiation, right?2010-10-27
  • 0
    @muntoo, yes, I have seen $f^{iv}$ for the derivative, but it has usually been with lower case letters as I have shown. Dunno why. But I have also seen numbers. I think PPJ is right in the comment below that the numbers are usually in parentheses.2010-10-27
  • 0
    @ross Is there any way to tell the difference between $sin^2(x)$ being $sin(sin(x))$ and $(sin(x))^2$?2010-10-27
  • 0
    @muntoo, The rule I said earlier "You can write anything you like" also applies to other people. Indeed, they write any nonsense they like without trying to be consistent or anything.2010-10-27
  • 0
    @muad OK, but is there anyway for me to write it in a way that will help others? (Besides $(sin(x))^2$)2010-10-27
  • 0
    Most commonly $sin^2(x)$ is $(sin(x))^2$ just because that shows up so often.2010-10-27
  • 0
    @ross I must not be being clear... what if $sin$ was replaced by $f$ or $ln$?2010-10-27
  • 3
    @muntoo then it gets dicier. log gets raised to powers, particularly in analysis of computing efficiency, so $log^n$ (though less likely ln) is probably a power. Other functions less so. You just need to read the definition and be careful. Often you can tell from the context, as if we are iterating it will all be about that.2010-10-28
  • 0
    One think about $f^4$ for the iteration is that it is consistent with $f^{-1}$ for the inverse. This is problematic with $\sin^{-1}$2010-10-28
0

You should define it this way:

$$ \begin{eqnarray} \text{iterate}_0(f) &:=& id \\ \text{iterate}_{n + 1}(f) &:=& \text{iterate}_{n}(f) \circ f \end{eqnarray} $$

Then write $\text{iterate}_4(f)(x)$.

  • 1
    what is the reason for downvoting this please? I'm interested in hearing it if you have some disagreement with this.2010-10-27
  • 1
    @muad I don't see anything wrong with this. Another case where you have to define your terms2010-10-27
  • 0
    So... what's $id$ supposed to be?2010-10-27
  • 1
    id is the identity function, that is, it maps its input to its output. id is f(x)=x for R to R functions, and f(x,y)=(x,y) for R^2 to R^2 functions and so on.2010-10-28
0

See "function powers" in Wikipedia "Function composition".

  • 2
    And, incidently, the power notation, $f^4(x)$ is different from the fourth derivative which is $f^{(4)}(x)$, so there shouldn't be any confusion.2010-10-27
  • 3
    This type of remark (just a pointer to wikipedia or similar) is best as a comment.2010-10-27