Preimage
The preimage is the original figure before a transformation is applied.
Definition
The preimage is the original shape before any transformation is applied; after the transformation, the resulting shape is called the image, denoted without prime marks ($A$, $B$, $C$) while the image uses prime marks ($A'$, $B'$, $C'$). Given the image and the transformation, you can work backwards to find the preimage by applying the inverse transformation: for a translation by $(a,b)$, the inverse is a translation by $(-a,-b)$, and for a dilation by $k$, the inverse dilates by $1/k$. Formally, the preimage of a set $S$ under transformation $T$ is $T^{-1}(S) = \{x : T(x) \in S\}$; for a bijective transformation, $T^{-1}(T(S)) = S$, and in topology, continuity of $T$ is defined by requiring preimages of open sets to be open, connecting the preimage concept to continuity.
Example
If triangle $ABC$ is rotated $90$ degrees to produce triangle $A'B'C'$, then $ABC$ is the preimage and $A'B'C'$ is the image. A dilation with scale factor $3$ from the origin maps preimage triangle $(1,1),(2,1),(1,3)$ to image $(3,3),(6,3),(3,9)$; to find the preimage given the image, divide coordinates by $3$. The preimage of the y-axis $\{(0,y)\}$ under $T(x,y) = (x^2, y)$ is $T^{-1}(\{(0,y)\}) = \{(x,y) : x^2 = 0\}$, which is the y-axis itself; for non-injective maps, preimages can be larger sets.
Key Insight
"Pre" means before: the preimage comes first, then the transformation is applied, giving the image, the original and the copy after the rule is applied. The topological definition of continuity via preimages unifies the epsilon-delta definition for real functions with continuity in abstract spaces, an abstraction that enabled the development of general topology and functional analysis in the early 20th century.