Let $S$ be the smallest set of functions from $\mathbb{R} \mapsto \mathbb{R}$ such that

$$\forall n \in \mathbb{N}, (\alpha, \beta) \in \mathbb{R^2}, f \in S \implies (\alpha f + \beta)^n \in S~\wedge~e^f \in S$$

Does $S$ contain a function $f$ such that

$$\forall x \in \mathbb{R},\\0<f(x)<1\\\wedge~f(x)+f(-x)=1\\\wedge~f'(x) > 0\\\wedge~f''(x) (f(x)-f(0)) > 0$$

(i.e. a sigmoid) ?

$S$ is of interest because for $f$ in $S$ the space $T(f) = \{x \mapsto f(x-\alpha)^\beta,\alpha>0,\beta \in \mathbb{R}\}$ can easily be parametrized with a finite number of parameters.