Notation for image and preimage Let $X$, $Y$ be sets and $f:X\rightarrow Y$ be a map.  Denoting the image of $D\subset X$ under $f$ by $f(D)$ can sometimes be confusing. As for preimages, I've seen unambiguous notation like $f^*\mathcal{O}$, where $\mathcal{O} \subset \mathcal{P}(Y)$. (This is also an example of the "confusing" notation of an image, though). For images, is analogous notation $f_*D$ for denoting the image of $D\subset X$ under $f$ used in the literature?
 A: There are a number of notations for the image, Lawvere and Rosebrugh list five different notations in their book "Sets for Mathematics", pg 137:
$$ \mathcal{P}f,\ f_!,\ f[\ ],\ \exists_f,\ \textrm{im}_f $$
The asterisk notation can be explained through topos theory.  The function $f:X \to Y$ gives rise to a geometric morphism $f: \mathbf{Set}/X \to \mathbf{Set}/Y$.  Every geometric morphism $f:\mathcal{E} \to \mathcal{F}$ contains an adjunction $(f^* \dashv f_*)$, called the inverse image and direct image respectively.  Some geometric morphisms have extra adjoints, the notation goes:
$$ f_! \dashv f^* \dashv f_* \dashv f^! $$
Functors with an exclamation mark (!), usually called shriek functors, are not present in every geometric morphism.  Subscript functors follow the geometric morphism, so $f_!, f_*: \mathcal{E} \to \mathcal{F}$, while superscript functors go in the opposite direction, $f^*, f^!: \mathcal{F} \to \mathcal{E}$.
If we return to a set function $f:X \to Y$, the corresponding geometric morphism $f$ has an inverse image $f^*$, which takes the pullback along $f$,  and direct image $f_* = \Pi_f$, which is a bit harder to describe.  But there is an extra adjoint, $f_! = \Sigma_f$, which composes with $f$.  This adjoint, when restricted to subobjects of the terminal object, corresponds to the image, not $\Pi_f$.
Following this convention the notation $f_*$ should not be used for the image, as the image is left-adjoint to the inverse image, as Zhen Lin says above.   However there is a functor $f_*$, which on subsets has the definition
$$ f_*(U) = \{ y \in Y\ |\ \forall x \in X, f(x) = y \to x \in U \} $$
This does not get as much use as the image.  I would still use $f_!$ for the image.  Whatever notation you use make it explicit what you mean, as not everyone will recognise this notation.
A: In my neck of the woods a common notation for $\{f(x)\mid x\in D\}$ is $f[D]$, and in some places you can also find people using the analysts-confusing double-prime, that is $f''D$.
For the preimage, the principle is the same: $\{x\mid f(x)\in O\}=f^{-1}[O]$.
When we teach this in the introductory course we say that if $D=\{x\}$ then we write $f[x]$ instead of $f[\{x\}]$ and similarly $f^{-1}[y]$ instead of $f^{-1}[\{y\}]$. The brackets remain to distinguish sets from elements.
A: Mac Lane and Birkoff in their Algebra use $f^*$ and $f_*$. Categorical notions permeate this book and the notation is consistent with the two functors image and inverse image on the category of sets.
