In set theory, a function often is its graph
Plot a function of one real variable and you usually get a curve of Cartesian pairs; two variables yield triples that can form a surface. Science and finance use such graphs constantly. Modern foundations often identify a function with that set of input-output pairs—while still tracking domain and codomain for words like surjective.
Formally, for a function from a domain X to a codomain Y, the graph is the collection of all pairs made of an input x and its output f of x. That collection sits inside the Cartesian product of X and Y. Strictly speaking a function is the triple of domain, codomain and graph, but set theorists routinely treat the function and its graph as one object, and the two words simply reflect different viewpoints on it.
A tiny example shows what the pairs can and cannot tell you. Take a function from the set 1, 2, 3 into the letters a, b, c, d that sends 1 to a, 2 to d and 3 to c. Its graph is just the three pairs 1 with a, 2 with d and 3 with c. Reading off first components recovers the domain exactly, and second components give the letters actually reached, a, c and d. The full codomain, which also includes b, cannot be deduced from those pairs, which is why claims about a function being onto need the codomain stated separately.
On the real line, the cubic x cubed minus 9x produces a set of pairs that, drawn on a Cartesian plane, traces a wavy curve. For two inputs, the rule sine of x squared times cosine of y squared generates triples that form a rippled surface in three-dimensional coordinates. Pictures of this kind are usually called plots, and in the simplest case one quantity is drawn against another on rectangular axes.
Richer drawings add the gradient of the function and several level curves, either painted onto the surface or projected down onto the base plane. One standard illustration of this treatment uses the function equal to minus the square of cosine x squared plus cosine y squared.
Source: Graph of a function
