 Graph of a function

For a function $f:A\to B$, its graph of a function is the subset $\{(x,f(x)):x\in A\}$ of the Cartesian product $A\times B$. Exactly one ordered pair has first coordinate $x$ for each $x\in A$. For example, the quadratic function $f(x)=x^2$ on the real line has the parabola as its graph.

The graph determines the values and the domain of a function, but alone does not record a larger specified codomain: the same set of ordered pairs can be regarded as lying in $A\times B$ or $A\times B^{\prime }$ when both contain all its outputs. It also differs from a graph in graph theory, whose edges need not specify exactly one output per input.

The graph of a function is a visual representation of the relationship between the inputs (independent variables) and outputs (dependent variables) of that function. In coordinate geometry, a function can often be represented in a two-dimensional space using a Cartesian coordinate system, where the x-axis represents the independent variable (often denoted as \( x \)) and the y-axis represents the dependent variable (often denoted as \( f(x) \) or \( y \)).