---
tag: 0Q3F20XB0A
kind: definition
name: Image of a set
msc: ["03E20"]
defines: 07XF282NWH
uses: [0XHDTE5YMZ, 0FE8Q5S42Y]
sources:
  - url: https://en.wikipedia.org/wiki/Image_(mathematics)
    locator: Image of a subset
---

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]] and let $S$ be a [[subset|0FE8Q5S42Y]] of $A$.
The image of $S$ under $f$ is the set
$$f(S) = \{f(s) \mid s \in S\} = \{b \in B \mid \exists s \in S, \, f(s) = b\}.$$

## Remarks

Another notation is $f[S]$. It avoids a confusion with $f(a)$ for an element $a$.
The image $f(A)$ of the whole domain is the [[range|0E9GTAV478]] of $f$.
Example: for $f(x) = x^2$ on $\mathbb{R}$, $f([-1, 2]) = [0, 4]$.
The image of a union is the union of the images, but the image of an intersection can be smaller. See [[image of an intersection|0RX7DDBYCK]].

## Explanations

### L1 human

To compute $f(S)$, apply $f$ to each element of $S$ and collect the results.
For $f(x) = x + 1$ and $S = \{1, 2, 3\}$, the image is $\{2, 3, 4\}$.