Preimage
Geometry & MeasurementThe 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. The preimage and image are related by the transformation rule.
Example
If triangle $ABC$ is rotated $90$ degrees to produce triangle $A'B'C'$, then triangle $ABC$ is the preimage and triangle $A'B'C'$ is the image. "Pre" means before, so preimage means "before the transformation."
Key Insight
"Pre" means before. The preimage comes first, then the transformation is applied, giving the image. In everyday language: the preimage is the original, the image is the copy after the rule is applied.
Definition
The preimage is the set of all original points of a figure before a transformation $T$ is applied. Each point $P$ in the preimage maps to an image point $P' = T(P)$. The preimage is denoted without prime marks ($A$, $B$, $C$) while the image uses prime marks ($A'$, $B'$, $C'$).
Example
Dilation with scale factor $3$ from 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 and transformation, apply the inverse: divide image coordinates by $3$ to get preimage.
Key Insight
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 translate by $(-a,-b)$. For a dilation by $k$, the inverse dilates by $1/k$.
Definition
The preimage of a set $S$ under transformation $T$ is $T^{-1}(S) = \{x : T(x) \in S\}$. For a bijective transformation, the preimage of the image is the original set: $T^{-1}(T(S)) = S$. In topology, continuity of $T$ is defined by requiring preimages of open sets to be open, connecting the preimage concept to continuity.
Example
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
The topological definition of continuity via preimages unifies the epsilon-delta definition for real functions with continuity in abstract spaces. This abstraction enabled the development of general topology and functional analysis in the early 20th century.