---
tag: 0Y0KFHF47G
kind: definition
name: Graph of a function
msc: ["03E20"]
defines: 0JBJYT69ZV
uses: [0XHDTE5YMZ, 0C81YBVTWH]
sources:
  - url: https://en.wikipedia.org/wiki/Graph_of_a_function
    locator: lead section
---

## Definition

Let $f \colon A \to B$ be a [[function|0XHDTE5YMZ]].
The graph of $f$ is the set
$$\Gamma(f) = \{(a, f(a)) \mid a \in A\} \subseteq A \times B.$$

## Remarks

For a real function $f \colon \mathbb{R} \to \mathbb{R}$, the graph is a curve in the plane $\mathbb{R}^2$.
Let $G \subseteq A \times B$.
$G$ is the graph of a function from $A$ to $B$ if and only if each $a \in A$ is the first entry of exactly one pair.
In the set-theoretic [[definition of a function|0XHDTE5YMZ]], the function and its graph are the same set.

## Explanations

### L1 human

The graph of $f(x) = x^2$ is the parabola $\{(x, x^2) \mid x \in \mathbb{R}\}$.
Each vertical line meets it in exactly one point. This is the vertical line test.