0-29
SECTION 0.3
.
.
Inverse Functions
29
(a)
f
(0
.
4)
,
(b)
f
(0
.
39)
,
(c)
f
(1
.
17)
,
(d)
f
(1
.
20)
,
(e)
f
(1.8),
(f)
f
(1.81). Repeat (c)–(d) if the graphing window is zoomed in
so that
x
=
1
.
00
,
1
.
01
,...,
1
.
20 and
y
=
1
.
30
,
1
.
31
1
.
50
.
Repeat (e)–(f) if the graphing window is zoomed in so that
x
=
1
.
800
,
1
.
801
1
.
820 and
y
=
3
.
200
,
3
.
205
3
.
300
.
2.
Graph
y
=
x
2
−
1
,
y
=
x
2
+
x
−
1
,
y
=
x
2
+
2
x
−
1
,
y
=
x
2
−
x
−
1
,
y
=
x
2
−
2
x
−
1 and other functions of the
form
y
=
x
2
+
cx
−
1
.
Describe the effect(s) a change in
c
has
on the graph.
3.
Figures 0.31 and 0.32 provide a catalog of the possible
types of graphs of cubic polynomials. In this exercise, you
will compile a catalog of graphs of fourth-order polyno-
mials (i.e.,
y
=
ax
4
+
bx
3
+
2
+
dx
+
e
)
.
Start by using
your calculator or computer to sketch graphs with different
values of
a
,
b
,
c
,
d
and
e
.
Try
y
=
x
4
,
y
=
2
x
4
,
y
=−
2
x
4
,
y
=
x
4
+
x
3
,
y
=
x
4
+
2
x
3
,
y
=
x
4
−
2
x
3
,
y
=
x
4
+
x
2
,
y
=
x
4
−
x
2
,
y
=
x
4
−
2
x
2
,
y
=
x
4
+
x
,
y
=
x
4
−
x
and so on.
Tryto determine what effect each constant has.
f
g
x
Domain
{
f
}
y
Range
{
f
}
FIGURE 0.41
g
(
x
)
=
f
−
1
(
x
)
y
x
8
4
2
±
2
±
4
6
2
1
±
2
y
²
x
3
FIGURE 0.42
Finding the
x
-value corresponding
to
y
=
8
0.3
INVERSE FUNCTIONS
The notion of an
inverse
relationship is basic to many areas of science. The number of
common inverse problems is immense. As only one example, take the case of the electro-
cardiogram (EKG). In an EKG, technicians connect a series of electrodes to a patient’s chest
and use measurements of electrical activity on the surface of the body to infer something
about the electrical activity on the surface of the heart. This is referred to as an
inverse
prob-
lem, since physicians are attempting to determine what
inputs
(i.e., the electrical activity
on the surface of the heart) cause an observed
output
(the measured electrical activity on
the surface of the chest).
The mathematical notion of inverse is much the same as that just described. Given an
output (in this case, a value in the range of a given function), we wish to ﬁnd the input (the
value in the domain) that produced that output. That is, given a
y
∈
Range
{
f
}
,
ﬁnd the
x
∈
Domain
{
f
}
for which
y
=
f
(
x
)
.
(See the illustration of the inverse function
g
shown in
Figure 0.41.)
For instance, suppose that
f
(
x
)
=
x
3
and
y
=
8
.
Can you ﬁnd an
x
such that
x
3
=
8?
That is, can you ﬁnd the
x
-value corresponding to
y
=
8? (See Figure 0.42.) Of course, the
solution of this particular equation is
x
=
3
√
8
=
2
.
In general, if
x
3
=
y
,
then
x
=
3
√
y
.In
light of this, we say that the cube root function is the
inverse
of
f
(
x
)
=
x
3
.
EXAMPLE 3.1
Two Functions That Reverse the Action of Each Other
If
f
(
x
)
=
x
3
and
g
(
x
)
=
x
1
/
3
,
show that
f
(
g
(
x
))
=
x
and
g
(
f
(
x
))
=
x
,
for all
x
.
Solution
For all real numbers
x
,
we have
f
(
g
(
x
))
=
f
(
x
1
/
3
)
=
(
x
1
/
3
)
3
=
x
and
g
(
f
(
x
))
=
g
(
x
3
)
=
(
x
3
)
1
/
3
=
x
.
±
Notice in example 3.1 that the action of
f
undoes the action of
g
and vice versa. We
take this as the deﬁnition of an inverse function. (Again, think of Figure 0.41.)