(5/31/07)
Section 0.4
Inverse functions and logarithms
Overview:
Some applications require not only a function that converts a number
x
into a number
y
, but
also its
inverse
, which converts
y
back into
x
. In this section we analyze inverse functions and discuss
logarithms
, which are the inverses of exponential functions.
Topics:
•
Inverse functions
•
Changing variables
•
Restricting domains
•
Logarithms
•
The common and natural logarithms
•
Laws of logarithms
Inverse functions
The function of Figure 1 converts the year
t
into the number
N
=
f
(
t
) (millions) of cars that were
registered in California at that time.
(1)
For example, starting with
t
= 1980 on the horizontal
t
axis,
moving vertically to the graph, and then horizontally to the vertical
N
axis, we obtain
N
=
f
(1980) =
17
.
4. At the beginning of 1980 there were 17.4 million cars registered in California.
t
N
(millions)
10
20
30
17
.
4
N
=
f
(
t
)
1940
1960
1980
2000
N
10
17
.
4
30
t
(year)
1940
1960
1980
2000
t
=
f
–1
(
N
)
(millions)
FIGURE 1
FIGURE 2
The
inverse
of
f
, denoted
f
–1
and read “
f
inverse,” has the opposite effect.
†
It converts the
number
N
into the time
t
=
f
–1
(
N
) when there were
N
cars registered. We could use Figure 1 to study
f
–1
by having its variable (the independent variable) be on the vertical
N
axis and its values on the
horizontal
t
axis. For example, we could start with
N
= 17
.
4 on the vertical axis, move horizontally to
the graph, and then vertically to the horizontal
t
axis to obtain
t
= 1980. This gives
f
–1
(17
.
4) = 1980,
which means that number of cars registered was 17.4 million at the beginning of 1980.
(1)
Data adapted from
CALPERG Citizen Agenda
, Vol. 18, No. 2, Los Angeles: CALPERG, 2000, p. 5.
†
Because the symbol
f

1
also denotes the reciprocal 1
/f
of
f
, its meaning must be determined from the context in which
it is used.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
p. 2 (5/31/07)
Section 0.4, Inverse functions and logarithms
This procedure for studying the inverse is, however, not convenient because we prefer to have the
independent variable on the horizontal axis. To achieve this, we flip the drawing about the diagonal line
that makes a 45
◦
degree angle with the positive
x
 and
y
axes. This yields the graph
t
=
f
–1
(
N
) of
the inverse function in Figure 2, where the variable
N
is on the horizontal axis and the values
t
of the
function are on the vertical axis. The graph of
f
–1
in Figure 2 is the mirror image of the graph of
f
in
Figure 1 with respect to the diagonal line.
Here is a general definition:
Definition 1
Suppose that
y
=
f
(
x
)
is a function such that for each
y
in its range, the equation
y
=
f
(
x
)
has one and only one solution
x
. Then
x
is the value of the
inverse function
x
=
f
–1
(
y
)
at
y
. Thus,
†
x
=
f
–1
(
y
)
⇐⇒
y
=
f
(
x
)
.
(1)
Question 1
Suppose that
f
has an inverse
f
–1
, that
f
(1) = 20, and that
f
–1
(30) = 2. What are
the values of
f
–1
(20) and
f
(2)?
This is the end of the preview.
Sign up
to
access the rest of the document.
 Summer '08
 Eggers
 Math, Inverse Functions, Natural logarithm, Logarithm

Click to edit the document details