Is there a short way to say $f(f(f(f(x))))$?
I know you can use recursion:
$g(x,y)=\begin{cases} f(g(x,y-1)) & \text{if } y > 0, \ \newline x & \text{if } y = 0. \end{cases}$
Is there a short way to say $f(f(f(f(x))))$?
I know you can use recursion:
$g(x,y)=\begin{cases} f(g(x,y-1)) & \text{if } y > 0, \ \newline x & \text{if } y = 0. \end{cases}$
I personally prefer $f^{\circ n} = f \circ f^{\circ n-1} = \dotsb = \kern{-2em}\underbrace{f \circ \dotsb \circ f}_{n-1\text{ function compositions}}$
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 $x\le 0$
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 $\text{iterate}_4(f)(x)$.
See "function powers" in Wikipedia "Function composition".