---
tag: 0E9GTAV478
kind: definition
name: Range of a function
msc: ["03E20"]
defines: 03FHD14RKC
uses: [0XHDTE5YMZ]
sources:
  - url: https://en.wikipedia.org/wiki/Range_of_a_function
    locator: lead section
---

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The range of $f$ is the set
$$\operatorname{ran}(f) = \{f(a) \mid a \in A\} = \{b \in B \mid \exists a \in A, \, f(a) = b\}.$$

## Remarks

The range is also called the image of $f$, and it is the [[image|0Q3F20XB0A]] $f(A)$ of the whole domain.
Some authors use "range" for the [[codomain|0F9JHP8AJN]]. This node uses the meaning above.
For $f \colon \mathbb{R} \to \mathbb{R}$ with $f(x) = x^2$, the range is $[0, \infty)$.

## Explanations

### L1 human

The range is the set of outputs that actually occur.
It can be smaller than the codomain.