Transcript 1.6 – Inverse Functions and Logarithms
1.6 – Inverse Functions and Logarithms
One-To-One Functions
A function is
one-to-one
if no two domain values correspond to the same range value.
( Algebraically, a function is
one-to-one
x
2 ) for all
x
1 ≠
x
2 .
if
f
(
x
1 ) ≠
f
Graphically, a function is one-to-one if its graph passes the
horizontal line test
. That is, if any horizontal line drawn through the graph of a function crosses more than once, not it is one-to-one .
Try This
Determine if the following functions are one-to-one.
(a)
f
(
x
) = 1 + 3
x
– 2
x
4 (b)
g
(
x
) = cos
x
+ 3
x
2 (c)
h
(
x
)
e x
e
x
2 (d)
f
(
x
) 5
x
Inverse Functions
The
inverse of a one-to-one
function is obtained by exchanging the domain and range of the function. The inverse of a one-to-one function
f
denoted with
f
-1 .
is Domain of Range of
f f
= Range of
f
-1 = Domain of
f
-1
f
−1 (
x
) =
y
<=>
f
(
y
) =
x
Try This
Sketch a graph of
f
(
x
) = 2
x
and sketch a graph of its inverse. What is the domain and range of the inverse of
f
.
Domain: (0, ∞) Range: (-∞, ∞)
Inverse Functions
You can obtain the graph of the inverse of a one to-one function by reflecting the graph of the original function through the line
y
=
x
.
Inverse of a One-To-One Function
To obtain the formula for the inverse of a function, do the following: 1. Let
f
(
x
) =
y
.
2. Exchange
y
and
x
.
3. Solve for
y
.
4. Let
y
=
f
−1 (
x
).
Inverse Functions
Determine the formula for the inverse of the following one-to-one functions.
(a) (b)
f h
( (
x
)
x
) 2
x
3 3
x
3 1
x
2 (c)
g
(
x
) 3
x
Logarithmic Functions
The inverse of an exponential function is called a logarithmic function.
Definition:
x
=
a y
if and only if
y
= log
a x
Logarithmic Funcitons
The function
f
(
x
) = log
a
function.
x
is called a logarithmic
Domain:
(0, ∞)
Range:
(-∞, ∞)
Asymptote:
x
= 0
Increasing
for a > 1
Decreasing
for 0 < a < 1
Common Point:
(1, 0)
Logarithmic Functions
Now determine the inverse of
g
(
x
) = 3
x
.
Definition:
x
=
a y
if and only if
y
= log
a x g
1 (
x
) log 3
x
Properties of Logarithms
1. log
a 2. a a
log
a a x x
(
a x
) =
x
x x
for all
x
for all
x
> 0 3. log
a
(
xy
) = log
a x
4. log
a
(
x
/
y
) = log
a
5. log
a 6. log a x n
=
n
log
a a x = x x
+ log
a x y
– log
a y
Common Logarithm
: log 10
Natural Logarithm
: log
e x x
= log = ln
x x All the above properties hold.
Properties of Logarithms
The natural and common logarithms can be found on your calculator. Logarithms of other bases are not. You need the change of base formula.
log
a x
log
b x
log
b a
where
b
is any other appropriate base.
Try These
1. Determine the exact value of log 8 2. Determine the exact value of ln
e
2.
2.3
.
3. Evaluate log 7.3
5 to four decimal places.
4. Write as a single logarithm: ln
x
+ 2ln
y
– 3ln
z
.
5. Solve 2
x
+ 5 = 3 for
x
.
Try This
In the theory of relativity, the mass of a particle with velocity
v
is
m
f
(
v
)
m
0 1
v
2 /
c
2 where
m
0 is the mass of the particle and
c
is the speed of light in a vacuum. Find the inverse function of
f
and explain its meaning.