Give example(s) where all roots of $f(x)=f^{-1}(x)$ do not lie on $y= x$ or $y=-x$

Question

If $f:\mathbb R\to\mathbb R,f(x)=x|x|$ and $g: \mathbb R\to \mathbb R, g(x)=-x|x|$ then all three roots of $f(x)=f^{-1}(x)$ and $g(x)=g^{-1}(x)$ lie on $y=x$ and $y=-x$, respectively.

It will be interesting to have concrete example(s) of $h: A \to B$ where all roots of $ h(x)=h^{-1}(x)$ do not lie on $y= x$ or $y=-x$. Can you suggest one?

Edit Please do not consider the case(s) of no real rootroot and self inverse functions.

Answer 1

For sets $A=B=\mathbb R\setminus\{0,1\}$, $h:A\to B$,

$$h(x)= \begin{cases} \frac 1x & x>0 \land x\ne 1\\ 2x & x< 0 \end{cases},$$

some and all roots of $h(x)=h^{-1}(x)$ do not lie on $y=x$ or $y=-x$.

$h$ is not a self-inverse function.

Answer 2

If $f(x)=f^{-1}(x)$ then I'd expect $f(f(x))=x$. By definition if $f(f(x))=x$ and $f(x) \ne x$ then $x$ is a periodic point for $f$ of prime period $2$.

There are many classes of functions that give solutions. For example suppose $g_\lambda(x)=\lambda \cdot x\cdot(1-x)$. What is the smallest real number for which $g_\lambda(x) =x$ has a solution in the reals? What about $g_\lambda(g_\lambda(x))=x$? Values of lambda in this example or $c$ in the other polynomial example are parameter values which determine whether there are fixed points, points of period two, period 4, and so on. I'd expect this to give you what you are looking for. Topics associated with this are Bifurcation Diagrams as well as the above mentioned Periodic Points.

$f(f(x))=x$

Let $ f(x)=x^2+c$

$f(f(x))= (x^2+c)^2+c = x^4+2cx^2+c^2+c$

$x^4+2cx^2 -x+c^2+c=0$

$(x^2-x+c)(Ax^2+Bx+C)=x^4+2cx^2-x+c^2+c$

$A=1. $

$B-A=0\implies B=1.$

$C-B+Ac=2c\implies C=c+1$

$(x^2-x+c)(x^2+x+c+1)$

$x=\frac{-1 \pm \sqrt{-3-4c}}{2}$

$f(x)=\frac{ -2-4c \mp 2\sqrt{-3-4c}}{4}= \frac{-1 \mp \sqrt{-3-4c}}{2}-c \ne x$

Answer 3

A continuous one piece function $f:(0,\infty)\to (0,\infty), f(x)=(0.01)^x$ is a bijection

and its inverse is $f^{-1}:(0,\infty)\to (0,\infty), f(x)=\log_{0.01} x$

See in the fig. below, the equation $f(x)=f^{-1}(x)$ has three roots all of them do not fall on $y=x$ or $y=-x$. Here, $f(x)$ is blue, $f^{-1}(x)$ is red and $y=x$ is green.