What is a function that satisfies the iterated functional equation $f(f(x))=-x$?

[Jameson]

I don't know what to title this but will change it if $f(f(x))$ has a name. Anyway, I need to find $f(x)$ such that $f(f(x))=-x$. My friend gave me this challenge question and I haven't been able to figure it out.

There are many examples where $f(f(x))=x$ for example \(\displaystyle f(x)=\frac{1}{x}\) but that doesn't work for this one. I also tried a piecewise function but that didn't work either.

Lastly $f: \mathbb{R} \rightarrow \mathbb{R}$

Any ideas?

 

[MarkFL]

How about:

$f(x)=ix$?

 

[Jameson]

How about:

$f(x)=ix$?

$f(2)=2i$, $f(2i)=2i^2=-2$
$f(-2)=-2i$, $f(-2i)=-2i^2=2$

\(\displaystyle f \left( \frac{1}{2} \right)= \frac{1}{2}i\), \(\displaystyle f \left( \frac{1}{2}i \right) \frac{1}{2}i^2=-\frac{1}{2}\)

Yup, seems to work. Nice! I wonder if there are any others as well.

 

[Fernando Revilla]

Nice! I wonder if there are any others as well.

At least, if $f:\mathbb{C}\to \mathbb{C}$ has de form $f(x)=ax+b$, necessarily $f(x)=ix$ or $f(x)=-ix$.

 

[Jameson]

Doh! I forgot my friend stipulated that f: R -> R.

 

[MarkFL]

For a function of the form:

$f(x)=kx^n$ then $f(f(x))=k^{n+1}x^{n^2}$

and the only solution I can find is for $n=1$ leading to the example I gave. (I missed the other example cited by Fernando Revilla which comes from $k^2=-1\,\therefore\,k=\pm i$)

I bet there are other forms that might work though. I have to run now, but I will be thinking about this.

 

[Fernando Revilla]

Doh! I forgot my friend stipulated that f: R -> R.

The trick is to consider a set of disjoint intervals $(0,1],(1,2],(2,3],(3,4],\ldots $ covering $\mathbb{R}^+$. We define $f:\mathbb{R}^+\to \mathbb{R}$ in the following way

\begin{align}
f(0) &= 0 \\
f(x) &= x+1 &&\mbox{if $x\in(2k-2,2k-1]$}\\
f(x) &= 1-x &&\mbox{if $x\in(2k-1,2k]$}\\
\end{align}

We easily verify $f(f(x))=-x$. Now we can extend $f$ to $\mathbb{R}$ by odd symmetry i.e. if $x<0$ we define $f(x)=-f(-x)$ and the condition $f(f(x))=-x$ is verified for all $x\in\mathbb{R}$.

 

[MarkFL]

Doh! I forgot my friend stipulated that f: R -> R.

Allow me to offer a bit of friendly advice when posting problems on math help sites:

When posting the problem, please include all relevant information. This ensures that you get the best and most timely help possible.

<ducking&running&lmao> (Rofl)(Wave)(Beer)

 

[Rashad9607]

Hello all,

I don't recall where we originally found the problem but we certainly had some fun with it before I offered it to Jameson.

A similar problem was posted on stackoverflow, of all places.

Here was my lamentable first solution:

$f(x)=\left\{\begin{matrix}
0 & | &x=0 \\
-\frac{sign(x)}{|x|+1}{} & | & x\in \mathbb{Z} \setminus \{0\} \\
\frac{1}{x}-sign(x) & | & \frac{1}{x}\in \mathbb{Z}\\
\frac{1}{x} & | & |x|>1, x\notin\mathbb{Z}, \frac{1}{x}\notin\mathbb{Z}\\
-\frac{1}{x} & | & |x|<1, \frac{1}{x}\notin\mathbb{Z}
\end{matrix}\right.$

There are three things going on:

1. 0 maps to 0 (I'll write 0 > 0), which it must do for all f
2. The integers and their reciprocals: 1>-1/2>-1>1/2>1> ...; 2>-1/3>-2>1/3>2>...
3. f maps everything else to its inverse or negative inverse: 3/5>-5/3>-3/5>5/3>3/5>...; pi>1/pi>-pi>-1/pi>pi>...

The pieces came from trying something and making different rules for the numbers it didn't work for.

After finding a few other solutions similar to Fernando's very smooth function, we wondered if all such f operated on these kinds of tricks, and the answer was an emphatic No. We could map pi^e > 2 > -pi^e > -2 > pi^e if we wanted, just as long as each nonzero pair (x,-x) was itself paired with another nonzero (y,-y), and those pairings of pairs could be whatever we liked. That thought led to the attached document. I hope you guys find it interesting and correct!

 

[Nono713]

So I came across this today:

calculus - Find a real function, $f$, s.t. $f(f(x)) = -x$. - Mathematics

Knew I already saw it somewhere... maybe someone will find the link useful :p