Roots of $f(x)=f^{-1}(x),~ f:\mathbb R \rightarrow \mathbb{R}$

Question

I find that roots of $$\tag{1} f(x)=f^{-1}(x),$$ where $f: \mathbb{R} \rightarrow \mathbb{R}$ are the same as those of $f(x)=x$ and $f^{-1}(x)=x$ if $f(x)$ is an increasing function. If $f(x)$ is decreasing then roots of (1) are the same as $f(x)=-x$ and $f^{-1}(x)=-x.$ Two functions: $f(x)= x^3$ and $f(x)=-x^3$ are examples.

In general, I would like to know if it is necessarily true, are there exceptions? What could be a generalization of roots of (1) in the light of these two cases.

Edit: As per the answer/response of Aditya below self inverse functions may be exceptional to this conjecture.

Edit: Can there be some generalization wherein the roots of $f(x)=f^{-1}(x)$ would be same as that of $f(x)=p(x)$ and $f^{-1}(x)=p(x)$, what could be $p(x)$ other than $x$?

Answer 1

what you have written is not correct as there can exist functions such that $f(x) = f^{-1}(x) \ \forall x \in \mathbb{R} $ a example $f(x) = \frac{1}{x}$

Answer 2

You are right only for increasing functions. For decreasing functions you are right only under an additional assumptions like oddness (see below). For instance, the function $f(x)=5-x$ is a decreasing bijection of $\mathbb R$ with $f(0)=5$ and $f(5)=0$ and thus $f(0)=f^{-1}(0)=5$ and $f(5)=f^{-1}(5)=0$.

For (necessarily strictly) increasing bijections of $\mathbb R$, however, you are right. The proof goes like this: If $f$ is strictly increasing and $f(x)=f^{-1}(x)=y$ then $f(y)=x$. But $f(x)=y$ and $f(y)=x$ together are for strictly increasing $f$ impossible in the cases $x<y$ or $y<x$. Hence, necessarily $x=y$, that is, it follows that $f(x)=x=f^{-1}(x)$.

You are also right for bijections of $\mathbb R$ which are (necessarily strictly) decreasing and odd (that is, $f(-x)=-f(x)$ for all $x$). Indeed, if $f(x)=f^{-1}(x)=y$, hence $f(y)=x$, then we have by the oddness the two equalities $f(x)=y$ and $f(-y)=-x$. These equalities imply in case $x<-y$ by the monotonicity that $y>-x$ and in case $x>-y$ by the monotonicity that $y<-x$. In both cases, one gets a contradiction by multiplying one of the inequalities with $-1$. Thus, necessarily $x=-y$, that is, it follows that $f(x)=-x=f^{-1}(x)$.

Answer 3

If you require a decreasing function but do not require it to be odd, in addition to linear functions from the class $x\mapsto a-x$ (for constant $a$) you also have functions like

$$ f(x) = \sqrt{3x^{2}+1}-2x. $$

The graph of $y = f(x)$ is one branch of a hyperbola with an axis along the line $y = x.$ It takes a bit of work to confirm this, but $f(f(x)) = x$ and therefore $f^{-1}(x) = f(x).$

Basically, the condition $f^{-1}(x) = f(x)$ requires that the function be symmetric in reflection across the line $y=x.$ Any further properties come from additional restrictions that you place.

With that in mind, you can take any differentiable function whose derivative always has magnitude less than $1$ and rotate its graph $45$ degrees clockwise to get a function for which $f^{-1}(x) = f(x).$ Expressing the new function in a form as nice as the original function is not always possible, however.