¤
Descartes ¤The idea of this system was developed in 1637
in writings by Descartes and independently by Pierre de Fermat.
Both authors used a single axis in their treatments and have
a variable length measured in reference to this axis. The
concept of using a pair of axes was introduced in 1649
by Frans van Schooten and his
students.
Choosing a Cartesian coordinate system for a straight
line—means choosing a point O of the line (the
origin), a unit of length, and an orientation for the line.
A line with a chosen Cartesian system is called a number
line.
Points in a Euclidean plane are located by an ordered
pair of cartesian coodinates, $(x,y)$.
A line in a cartesian/Euclidean plane is identified
with a set of points where a point is on the line if
and only of its cartesian coordinates $(x,y)$ satisfy an
"linear" equation: $Ax + By = C$.
Points in a Euclidean space are located by an ordered
triple of cartesian coodinates, $(x,y,z)$.
A plane in a cartesian/Euclidean space is identified
with a set of points where a point is on the plane if
and only of its cartesian coordinates $(x,y,z)$ satisfy an
"linear" equation: $Ax + By +Cz = D$.
Leibniz 
Dirichlet
Lobachevsky
¤
The
concept of "function" was coined by Gottfried
Leibniz, in a 1673 letter,
to describe a quantity related to a curve.
Peter Gustav
Lejeune Dirichlet (in about 1837)and Nikolai Lobachevsky are
traditionally credited with independently giving the modern
"formal" definition of a function as a relation
in which every first element has a unique second element.
A non-vertical cartesian line corresponds to a
"linear function" where $y = f(x) = mx+b = Ax + B$.
A non-vertical cartesian plane corresponds
to a "linear function" where $z = f(x,y) = Ax + By + C$.
Suppose we start with a linear function:
The tradition is to visualize a linear function using the
cartesian plane or space and the graph of the function is a
line in the plane or a plane in space.
Instead of using a graph to visualize (linear)
functions, an alternative is available:
Mapping Diagrams¤
For more on linearity
[composition and inverses] go to http://flashman.neocities.org/MD/section-2.1LF.html.
Mapping Diagrams from A(lgebra)
B(asics) to C(alculus) and D(ifferential) E(quation)s.
A Reference and Resource Book on Function Visualizations
Using Mapping Diagrams
http://flashman.neocities.org/MD/section-2.1LF.html