For certain choices of elementary function $f:\Bbb R\to\Bbb R$, the inequality
$$f(x)^2 + f(y)^2 \le 1$$
describes a region $D\subset\Bbb R^2$ that is bounded, connected, symmetric (i.e. containing at least one axis of symmetry, but not necessarily radially), and in some cases strongly resembles the unit disk. $D$ is the interior of what I think might be classified as a family of superformulae.
- $f(z)=z$ of course gives the unit disk itself
- $f(z)=z^n$ for $n\in \Bbb N$ traces out a progressively rectangular disk as $n$ gets larger
- $f(z)=\sqrt[2n+1]{z}$ for $n\in\Bbb N$ draws out a progressively "pinched" astroid (another case shown in the same Wiki link)
- $f\in\left\{\sin^{-1},\sinh^{-1},\tan^{-1},\tanh^{-1}\right\}$, but not $f=\cos(\operatorname h)^{-1}$ nor any of these functions' "reciprocal" companions, each generate a pseudo-circular region
- $f=\log$ produces a rounded triangular region (the 2D case of the surface discussed in this question)
I've played around with a few compositions of these functions (avoiding reciprocals or division), and some exotic combinations had preserved boundedness, though often lost connectedness, symmetry, and disk-similarity.
For the polynomial case, varying choices of coefficients and powers allow $D$ to be a disconnected region, which I assume is due to the nature of the poly's roots.
Question:
Without the help of CAS or graphing utilities, are there any special properties of compound functions $f$ that can help predict the general shape of $D$?
By "predicting the general shape", I mean: can we answer definitively without unsheathing a graphing calculator,
- Is $D$ bounded?
- Is $D$ connected?
- Is $D$ symmetric?
The polynomial case with, say $\operatorname{deg}f\le4$, strikes me as the simplest, so I have no problem with an answer that focuses on only this scenario.