If a one-to-one function's inverse is the same what must be true of the graph of f?

Question

As a followup to this question. I'm trying to determine what must be true of the graph of $f$ in these cases. I've examined the two functions $f(x)= x$ and $f(x)= \frac{1}{x}$ and I'm not seeing any unifying graphic truth. Is there a graphic truth to be found for one-to-one function's where the inverse happens to be the same?

Answer 1

Inversion of the graph corresponds to the operation $U(x,y)=(y,x)$. That is, the graph of the inverse of $f$ is $\{(f(x),x)\}$ (whereas the graph of $f$ is $\{(x,f(x))\}$). It follows that, in terms of graphs, inversion is reflection in the line $x=y$. To see this, consider the action of $U$ on the standard basis of $\mathbb R^2$ and compare to the action of reflection in the line $x=y$.

For the function $f(x)=1/x$, consider its graph. This is precisely its own reflection through the line $y=x$.

Answer 2

HINT

The graphs $y=\mathrm{f}(x)$ and $y=\mathrm{f}^{-1}(x)$ are symmetric about the line $y=x$.

That means, to go from the graph $y=\mathrm{f}(x)$ to the graph $y=\mathrm{f}^{-1}(x)$, you reflect in the line $y=x$.

If the graphs $y=\mathrm{f}(x)$ and $y=\mathrm{f}^{-1}(x)$ are the same then what does that tell you?