Graph of a function

Definition

Let $f \colon A \to B$ be a function. 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, 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.