Let $\mathcal R$ be any o-minimal expansion of the real ordered field $(\mathbb R,<,0,1,+,-,\cdot)$, and $\mathcal F$ be the class of functions (first-order) definable (with real parameters) in $\mathcal R$. On the one hand, $\mathcal F$ has various nice closure properties (in particular, it is closed under composition, taking inverse functions, and derivatives). On the other hand, o-minimality guarantees that for any $f\in\mathcal F$, $f\colon\mathbb R\to\mathbb R$, its positive set $\{x\in\mathbb R:f(x)>0\}$ is a finite union of points and intervals; in particular, $f$ is eventually positive, eventually negative, or eventually constant $0$.

Note that in practice, theories of structures known to be o-minimal are often also model complete, hence a function is definable iff its graph is a projection of a Boolean combination of positive sets of the basic functions included in its signature.

Wilkie proved that the exponential field $\mathbb R_{\exp}=(\mathbb R,\exp)$ is o-minimal. The class of functions definable in $\mathbb R_{\exp}$ includes the functions mentioned in your question, so the answer to your specific question is positive.

Even larger expansions of $\mathbb R$ are known to be o-minimal. First, by a result of van den Dries, $\mathbb R_\mathrm{an}$ is o-minimal, which is the expansion of $\mathbb R$ by all real-analytic functions $f\colon[0,1]^n\to\mathbb R$ (extended by the constant $0$ function outside $[0,1]^n$ to be defined on the whole of $\mathbb R^n$). Second, the pfaffian closure $\mathcal R_\mathrm{pfaff}$ of any o-minimal expansion $\mathcal R$ of $\mathbb R$ is again o-minimal, due to Speisseger. In particular, $\mathbb R_\mathrm{an,pfaff}$ is o-minimal. (The full definition of the pfaffian closure can be found e.g. in [1]. In particular, it includes all pfaffian functions such as $\exp$.)

[1] Patrick Speissegger, *Pfaffian Sets and O-minimality*, in: Lecture Notes on O-Minimal Structures and Real Analytic Geometry (C. Miller, J.-P. Rolin and P. Speissegger, eds.), Fields Institute Communications vol. 62, 2012, pp. 179–218, http://dx.doi.org/10.1007/978-1-4614-4042-0_5