Fix a vector $x\in\mathbb{R}^d$ and a smooth function $\phi:\mathbb{R}\rightarrow \mathbb{R}$. $\phi$ applies entrywise on vector inputs. Under what conditions we have the equality
$$\mathbb{R}^d=\text{span}(\{\phi(\alpha x)~\big |~\alpha\geq 0\})$$
Clearly if $\phi$ is the identity operator $\phi(x)=x$ or if any two entries of $x$ are equal, this equality does not hold. Main claim is equality holds for generic (i.e. almost all) $x,\phi$ choices.