where the solution may involve many steps, “doing it in your head” increases the
possibility of error, and may be (practically speaking) nearly impossible. One of
the tricks in
using
mathematics, rather than being used or pushed around by it, is
to be able to
translate
it back into common sense terms, whenever required.
Functional notation is a remarkable communication device. With just a few symbols,
it can provide a summary of the relation between an
infnite
number of values of one
variable and the corresponding values taken on by another variable. We simply need to
remember the function, and then just plug in the
specifc
values of one variable that we
are interested in and perform the operations on them that the function tells us to, to ±nd
the corresponding
specifc
values for the other variable. Functional notation thus saves
time, economizes on memory, and frees up our mind for
real
thinking. It can even some-
times suggest
new
ideas or connections and analogies that we hadn’t seen before, and
thus
improve
our thinking. We now brieﬂy review coordinates and graphs, and then
look at some
specifc
aspects of functions as they are used in economics.
1.2 GRAPHS AND COORDINATES
With a few exceptions (see Appendices 3 and 9 in the text, where you will ±nd some
3-dimensional
graphs), economists typically use
2-dimensional
graphs to depict function-
al relationships. Graphs have the advantage of giving us much qualitative information
regarding a functional relationship in
visual
form, so that we can grasp it “at a glance.”
Graphs also have some disadvantages: they are less precise than algebraic representa-
tions of functions, and on occasion they can be misleading. Yet on balance they are
extremely valuable aids to economic understanding.
MATH MODULE 1: FUNCTIONS, GRAPHS, AND THE COORDINATE SYSTEM
M1-3
y
x
xx
(0, 12)
(0, 8)
(4,0)
(8,0)
(12,0)
(0, 4)
0
A
(4, 8)
B
(8, 4)
(0, –)
(0, +)
(–, +)
(+, +)
(–, –)
(a)
(b)
(+, –)
(–, 0)
(+, 0)
H
C
FIGURE M.1-2