In mathematics, a pre-image refers to the input value(s) that map to a specific output value when a function is applied.

Claude 3.7 Sonnet Think 

---

In mathematics, a pre-image refers to the input value(s) that map to a specific output value when a function is applied.

More formally:

• If f: X → Y is a function and y ∈ Y is an element in the codomain
• The pre-image of y is the set of all elements x ∈ X such that f(x) = y

For a one-to-one (injective) function, each element in the codomain must have at most one pre-image in the domain. This means:

• If f(x₁) = f(x₂), then x₁ = x₂
• Or equivalently: different inputs must produce different outputs

In your example with f(x, y) = (sin(x), cos(y)):

• The element (0, 1) in the codomain has multiple pre-images
• Both (0, 0) and (2π, 0) in the domain map to (0, 1)
• Therefore, (0, 1) has at least two pre-images: (0, 0) and (2π, 0)

This violates the one-to-one property, which is why f is not one-to-one.

The pre-image concept is essential for understanding key function properties like injectivity (one-to-one) and surjectivity (onto), as well as for defining the inverse of a function.

---

https://www.reddit.com/r/explainlikeimfive/comments/5hzrug/eli5_what_are_images_and_preimages_of_sets/

From the explainlikeimfive community on Reddit

When you're talking about a specific function f(x), the image of a set S is just the result of applying f to everything in S. For example, if f(x) = x2 and S is the interval [0,2], you can say "the image of S under f is [0,4]", since squaring each real number between 0 and 2 will result in every real number between 0 and 4.

The preimage is just going in the other direction: given a set T, what input values, when plugged into f, produce outputs in T? For example, the preimage of [0,4] under f(x) = x2 is [-2,2], since squaring anything between -2 and +2 will give a result between 0 and 4 but squaring anything outside that range will result in something greater than 4.