In Figure M.1-5(a), all three curves are linear (straight-line) graphs of functions, with
y =
f(
x
) = b
+ mx,
where
b
is the vertical intercept and
m
is the slope. Only (1) and (3),
however, have
inverse
functions, with the form
x = g
(
y
) = (
y – b
)/
m.
In the case of the
horizontal (constant) function (2),
y = b
, with
m =
0, there is an in±nite number of val-
ues of
x
corresponding to the
only
value that
y
takes on (the entire range of
y
is
b
). Hence
if we pick a value of
x
, say
x
0
, then f(
x
0
) =
y
0
= b
, but there does not exist an
inverse
func-
tion that will get us
back
directly to
x
0
from
y
0
= b
.
In Figure M.1-5(b),
y = x
2
likewise has no inverse function, since every value of
y
in
the range of the function (0,
∞
) can be generated by
two
values of
x
:
y =
4 can be gener-
ated by
x
= +2 or by
x
= –2. If we restrict the domain of
x
to be non-negative, however,
then an inverse function,
x = g
(
y
) = |
Ï
y
w
| = |
y
0.5
|,
does
exist.
In Figure M.1-5(c), there is no function
y =
f(
x
), because each value of
x
generates
two
values for
y
, one positive and one negative. There
does
, however, exist a function,
x =
g
(
y
) =
y
2
, that assigns a value of
x
for each value of
y.
If we restrict the range of
y
to
non-
negative
values, then
y = f
(
x
) = |
Ï
x
w
| = |
x
0.5
|,
(M.1.8)
and the inverse function
x = g
(
y
) =
y
2
also exists.
Inverse functions have many uses in economics, but in this course one of their pri-
mary uses is with demand and supply functions, where we ±nd
both
expressions relat-
ing price and quantity useful, in different contexts. We need to be able to transform one
functional form into its inverse, when the inverse exists and is useful. As an exercise,
you may want to examine the Figures and equations
throughout
this Module, and iden-
tify which are proper functions, and
also
which functions have an inverse.
1.6 FUNCTIONS OF SEVERAL VARIABLES
Some of the most important functions you will study in the course are functions of
sev-
eral
variables. Economists assume, for instance, that a consumer’s preferences can be
represented by a
utility function
having the form
U = f
(
X
1
,
X
2
,
X
3
, .
..
, X
n
),
(M.1.9)
M1-8
MATH MODULE 1: FUNCTIONS, GRAPHS, AND THE COORDINATE SYSTEM
y
yy
y
=
f
(
x
)
=
b
+
mx
xx
b
(a)
=
(
)
=
2
=
2
x
=
2
(b)
=
g
(
)
=
2
(c)
m > 0
m = 0
m < 0
00
0
–
2+
2
4
1
2
3
FIGURE M.1-5