**Contents**:

## Preimage & Image in Geometry

In geometry, figures in a plane can be transformed in a variety of ways, including shifts and scaling, to produce new shapes. The new (transformed) shapes are called

**images**and the original, unaltered shapes are called

*preimages*. In other words, a transformation can be though of as a procedure that maps a preimage onto an image.

## Functions Preimage & Image

The idea of an image and a preimage when you’re working with functions is the same idea as translation a shape, only here you’re “translating” sets of numbers. If f(*a*) = *b*, then [1]:

*b*is the image of*a*under f (when the function is clear from the context, the “under*f*” part is often dropped).*a*is an element of the preimage of*b*.

Images are elements of the range and preimages are subsets of the domain. Therefore, there may be more than one preimage of *b*, but only one image of *a*. This is just an extension of what you probably already know about functions: they must be “one to one” or “many to one” but cannot be “one to many”.

## Properties of Images and Pre-Images

In the following list of properties, *f*: A → B indicates than *f* is function (or map) from A to B:

- If
*f*: A → B then*f*[A] = ∅*f*^{-1}[B] = ∅.

- If
*f*: A → B and X, Y ⊆ B then*f*^{-1}[X ∩ Y] =*f*^{-1}[X] ∩*f*^{-1}[Y]. - If
*f*: A → B and X, Y ⊆ B then*f*^{-1}[X ∪ Y] =*f*^{-1}[X] ∪*f*^{-1}[Y] - If
*f*: A → B and X, Y, ⊆ A then*f*[A ∪ B] =*f*|A| ∪*f*|B|*f*[A ∩ B] ⊆*f*|A| ∩*f*|B|

As a simple example, consider the function f(x) = √x:

- The image of 5 is √5.
- The preimage of 5 is 25; for ℕ(the set of natural numbers) it is the set of all perfect squares in ℕ [2].

## References

[1] Math 310 Functions Handout.

[2] Kirby, P. Introduction to Functions. Retrieved November 4, 2021 from: https://www.math.fsu.edu/~pkirby/mad2104/SlideShow/s1_2.pdf