Graph of a Function¶

Definition¶

Let $X$ be a subset of $\mathbb{R}\text{,}$ and let $f\colon X\to\mathbb{R}$ be a function. The graph of $f$ is

\{(x,f(x))\in\mathbb{R}^2\colon x\in X\},

which is a subset of the Cartesian product $X\times\mathbb{R}\subseteq\mathbb{R}^2\text{.}$

Interpretation¶

The graph turns a function into a picture. It collects the pairs $(x,f(x))$ as points of the plane, so that the behaviour of $f$ can be read off by eye: the Cartesian coordinate system places each input on the horizontal axis and its output on the vertical axis.

The graph is a set of points with exactly one point above each $x$ in the domain $X$ . So every vertical line $x=c$ meets the graph once when $c\in X$ and not at all when $c\notin X$ $X$ . This vertical line test decides which curves in the plane are graphs of functions of $x$ : the unit circle $x^2+y^2=1$ is not one, since the line $x=0$ meets it at $(0,1)$ and at $(0,-1)$ .

Horizontal lines describe outputs: the line $y=c$ meets the graph exactly above the inputs with $f(x)=c$ $c$ . In particular, the zeros of $f$ are where the graph meets the $x$ -axis, the range of $f$ is the projection of the graph onto the $y$ -axis, and $f$ is injective if and only if no horizontal line meets the graph more than once. Likewise, the graph of an increasing monotone function never falls from left to right.

A plot is a guide, not a proof. It shows finitely many sampled points, so a single missing point or very fast oscillation can be invisible, and a claim read off a picture still has to be checked from the formula. The idea returns later for functions of two variables, whose graphs are surfaces in space.

Key Results¶

Each input corresponds to exactly one point on the graph.
The graph encodes continuity and monotonicity visually.
Many analytical properties can be inferred from the graph.

Examples¶

The graph of $f(x)=x^2$ is a parabola.
The graph of $f(x)=\mathrm{e}^x$ grows rapidly.

Graphs of x, x squared, x cubed, 1/x, the square root of x and e to the x; the graph of 1/x has two branches because 0 is not in its domain.

The graph of $f$ is the set of points $(x,f(x))$ ; the graph of $f_4$ has two branches, as $0$ is not in its domain.

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 4

Aliases: function graph Symbols: element of, real numbers, subset

Last updated: 20 October 2026 03:02