= Solution
\b[Not <injective>; its <image of a function> is exactly $\{1,2,3,\ldots\}$.] For example, both $x=e^{-1}$ and $x=e^{-1/2}$ give the <ceiling function> value $1$. Since $0<x<1$, $(\ln x)^2>0$, so the value $0$ is never attained. For every positive <integer> $j$,
$$
f(x)=j\quad\Longleftrightarrow\quad e^{-\sqrt j}\leq x<e^{-\sqrt{j-1}}.
$$
Each such <interval> is nonempty and contained in $(0,1)$, with the upper endpoint $1$ excluded when $j=1$. Thus the answer concerning <surjectivity> depends on the convention for the <natural numbers>: \b[onto if $\mathbb N=\{1,2,\ldots\}$, and not onto if $0\in\mathbb N$]. The source does not settle that convention.
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