Graph_of_a_function

Definition

Given a mapping ${\displaystyle f:X\to Y,}$ in other words a function ${\displaystyle f}$ together with its domain ${\displaystyle X}$ and codomain ${\displaystyle Y,}$ the graph of the mapping is[4] the set

which is a subset of ${\displaystyle X\times Y}$. In the abstract definition of a function, ${\displaystyle G(f)}$ is actually equal to ${\displaystyle f.}$

One can observe that, if, ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m},}$ then the graph ${\displaystyle G(f)}$ is a subset of ${\displaystyle \mathbb {R} ^{n+m}}$ (strictly speaking it is ${\displaystyle \mathbb {R} ^{n}\times \mathbb {R} ^{m},}$ but one can embed it with the natural isomorphism).

Examples

Functions of one variable

The graph of the function ${\displaystyle f:\{1,2,3\}\to \{a,b,c,d\}}$ defined by

is the subset of the set ${\displaystyle \{1,2,3\}\times \{a,b,c,d\}}$

From the graph, the domain ${\displaystyle \{1,2,3\}}$ is recovered as the set of first component of each pair in the graph ${\displaystyle \{1,2,3\}=\{x:\ \exists y,{\text{ such that }}(x,y)\in G(f)\}}$. Similarly, the range can be recovered as ${\displaystyle \{a,c,d\}=\{y:\exists x,{\text{ such that }}(x,y)\in G(f)\}}$. The codomain ${\displaystyle \{a,b,c,d\}}$, however, cannot be determined from the graph alone.

The graph of the cubic polynomial on the real line

is

If this set is plotted on a Cartesian plane, the result is a curve (see figure).

Functions of two variables

The graph of the trigonometric function

is

If this set is plotted on a three dimensional Cartesian coordinate system, the result is a surface (see figure).

Oftentimes it is helpful to show with the graph, the gradient of the function and several level curves. The level curves can be mapped on the function surface or can be projected on the bottom plane. The second figure shows such a drawing of the graph of the function:

See also

References

• Zălinescu, Constantin (30 July 2002). Convex Analysis in General Vector Spaces. River Edge, N.J. London: World Scientific Publishing. ISBN 978-981-4488-15-0. MR 1921556. OCLC 285163112 – via Internet Archive.