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Second Semester Calculus, Math 155B
Inverse functions
:
A function
f
is onetoone (11)
on an interval if
2
1
2
1
)
(
)
(
x
x
x
f
x
f
=
⇒
=
for all values
1
x
and
2
x
in the interval.
The graph of a 11
function must pass the Horizontal Line Test
:
No horizontal line can intersect the graph in
more than one point.
If a function
f
is 11, it has an inverse function
defined by
y
x
f
x
y
f
=
=

)
(
if
only
and
if
)
(
1
.
This means that if
f
takes
x
to
y
, then
1

f
takes
y
to
x
.
The domain of
f
is the range of
1

f
, and the range of
f
is the domain of
1

f
.
A function and its inverse have the
following cancellation relations:
x
x
f
f
=

))
(
(
1
and
y
y
f
f
=

))
(
(
1
Exponential functions
:
If
n
is a positive integer and
a
is a positive real number, then
we know that
times
...
n
n
a
a
a
a
⋅
⋅
⋅
=
,
n
n
a
a
1
=

, and
1
0
=
a
.
For a rational number
q
p
r
=
with
0
q
,
q
p
q
p
r
a
a
a
=
=
.
If
x
is a real number, define
r
x
r
x
a
a
→
=
lim
for rational numbers
r
.
Properties of exponential functions
:
The exponential function
x
a
x
f
=
)
(
is a
continuous function with domain
)
,
(
∞
∞
and range
)
,
0
(
∞
.
If
x
and
y
are real numbers,
y
x
y
x
a
a
a
+
=
⋅
y
x
y
x
a
a
a

=
xy
y
x
a
a
=
)
(
x
x
x
b
a
ab
=
)
(
The number
e
:
The number
e
is defined to be that unique positive number with the
property that the slope of the tangent line to the graph of
x
e
y
=
is 1 when
x
= 0.
The
value of
e
is approximately 2.718.
The function
x
e
x
f
=
)
(
is called the natural
exponential function
.
This function is differentiable with
x
x
e
e
dx
d
=
)
(
and
(if u is a differentiable function of x)
dx
du
e
e
dx
d
u
u
=
)
(
The associated indefinite integral is
C
e
dx
e
x
x
+
=
∫
.
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View Full Document Logarithm functions
:
The general exponential function
x
a
x
f
=
)
(
is increasing if
1
a
and decreasing if
1
0
<
<
a
.
This means the function is 11 and has an inverse.
The inverse of the exponential function
x
a
x
f
=
)
(
is called the logarithm function to the
base
a
,
which is defined by
y
a
a
x
x
y
=
=
if
only
and
if
log
.
Since the
range
of the
exponential function is
)
,
0
(
∞
, the
domain
of the logarithm function is
)
,
0
(
∞
, which
means that the logarithm function is defined only for
positive
values of
x
.
Cancellation properties
:
y
a
y
a
=
log
and
x
a
x
a
=
log
Graphs of logarithm functions
:
If
1
a
, the logarithm function
x
y
a
log
=
is a
continuous, increasing function with domain
)
,
0
(
∞
and range
)
,
(
∞
∞
.
Natural logarithm
:
The logarithm function to the base
e
is called the natural logarithm
function
:
x
x
e
log
ln
=
.
The natural logarithm function is related to the general logarithm
function by the formula
a
x
x
a
ln
ln
log
=
.
If
x
and
y
are positive numbers and
r
is any real
number, then
y
x
xy
ln
ln
)
ln(
+
=
y
x
y
x
ln
ln
ln

=
(
29
x
r
x
r
ln
ln
=
The natural logarithm function and the natural exponential functions are inverses of each
other, so
x
e
x
=
)
ln(
for all
x
and
x
e
x
=
ln
for
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This note was uploaded on 02/27/2009 for the course MATH 155b taught by Professor Staff during the Fall '08 term at Vanderbilt.
 Fall '08
 Staff
 Inverse Functions

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