ap calculus ab chapter 1: prerequisites for calculus section 1.5: functions and logarithms
TRANSCRIPT
One-to-One Functions Inverses Finding Inverses Logarithmic Functions Properties of Logarithms Applications
…and why
Logarithmic functions are used in many applications including finding time in investment problems.
What you’ll learn about…
One-to-One Functions A function is a rule that assigns a single value in
its range to each point in its domain. Some functions assign the same output to more
than one input. Other functions never output a given value more
than once. If each output value of a function is associated
with exactly one input value, the function is one-to-one.
One-to-One Functions
( ) ( ) ( )A function is on a domain if
whenever .
f x D f a f b
a b
¹
¹
one - to - one
One-to-One Functions ( )
The horizontal line test states that the graph of a one-to-one function
can intersect any horizontal line at most once.
If it intersects such a line more than once it assumes the sam
y f x=
e -value
more than once and is not a one-to-one function.
y
Inverses Since each output of a one-to-one function comes
from just one input, a one-to-one function can be reversed to send outputs back to the inputs from which they came.
The function defined by reversing a one-to-one function f is the inverse of f.
Composing a function with its inverse in either order sends each output back to the input from which it came.
Slide 1- 7
Inverses
( )( )
( )( ) ( )( )
1
1 1
The symbol for the inverse of is , read " inverse."
1The -1 in is not an exponent; does not mean
If , then and are inverses of one another;
otherwise they are not.
f f f
f f xf x
f g x g f x f g
-
- -
=o o
Identity Function The result of composing a function and its inverse in
either orderis the identity function.
Example Inverses( ) ( ) 2Determine via composition if and , 0
are inverses.
f x x g x x x= = ³
( )( ) ( )
( )( ) ( ) ( )
22
2
x
f g x f x x x
g x g x x
x
f
= = = =
= = =
o
o
Writing f -1as a Function of x.
( )
( )1
Solve the equation for in terms of .
Interchange and . The resulting formula
will be .
y f x x y
x y
y f x-
=
=
Section 1.5 – Functions and Logarithms A function is an inverse of another function
if and only if
for all x in the domain.
To find the inverse of a one-to one function:1. Interchange x and y.2. Solve for y.
xf 1
xf
xxffxxff 11 and
Example Finding InversesGiven that 4 12 is one-to-one, find its inverse.
Graph the function and its inverse.
y x= -
Solve the equation for in terms of .
13
4Interchange and .
13
4
x y
x y
x y
y x
= +
= +
( ) 4 12f x x= -
( )1 13
4f x x- = +
[-10,10] by [-15, 8]
Notice the symmetry about the line y x=
Section 1.5 – Functions and Logarithms Ex:
1 1 1
1
Find and verify that .
3 3
2 23
2
32 2
2
2 3
2 3
1 2 3
2 3
1
f f f x f f x x
x xf x y
x xy
xy
yx y y
y
xy x y
xy y x
y x x
xy f x
x
Section 1.5 – Functions and Logarithms
You try: Show that the function
is one-to-one and find its inverse function. 3 6f x x
Section 1.5 – Functions and Logarithms To graph the inverse parametrically:
If is a function that can be defined parametrically as
then the inverse can be defined as
xfy
tfytx and xf 1
tytfx and
Base a Logarithmic Function
( )
( )
( )
The base logarithm function log is the inverse of
the base exponential function 0, 1 .
The domain of log is 0, , the range of .
The range of log is , , the domain of .
a
x
xa
xa
a y x
a y a a a
x a
x a
=
= ¹
¥
- ¥ ¥
Logarithmic Functions Logarithms with base e and base 10 are so important in
applications that calculators have special keys for them.
They also have their own special notations and names.
log ln is called the .
log log is often called the .10
y x xe
y x x
natural logarithm function
common logarithm function
= =
= =
Inverse Properties for ax and loga x
log
ln
Base : , 1, 0
Base : , ln , 0
a x
x x
a a x a x
e e x e x x
= >
= =
Properties of Logarithms
For any real numbers 0 and 0,
log log log
log log log
log log
x y
Product Rule : xy x ya a a
xQuotient Rule : x ya a ay
yPower Rule : x y xa a
= +
= -
=
Example Properties of Logarithms
Solve the following for .
2 12x
x
=
Take logarithms of both sides
PowerRule
2 12
ln 2 ln12
ln 2 ln12
ln12 2.3025853.32193
ln 2 .693147
x
x
x
x
=
=
=
= = »
Example Properties of Logarithms Solve the following for .
5 60x
x
e + =
Subtract 5
Take logarithm of both sides
5 60
55
ln ln55
ln55 4.007333
x
x
x
e
e
e
x
+ =
=
=
= »
Slide 1- 23
Change of Base Formula
lnlog
lna
xx
a=
This formula allows us to evaluate log for any
base 0, 1, and to obtain its graph using the
natural logarithm function on our grapher.
a x
a a ¹
Slide 1- 24
Example Population Growth0.015The population of a city is given by 105,300
where 0 represents 1990. According to this model,
when will the population reach 150,000?
tP P e
t
=
=
0.015
0.015
0.015
0.015
Solve for t
Take logarithm of both sides
Inverse property
105,300 , 150,000
105,300
150,000
105,300
150,000ln ln
105,300
0.353822= 0.01
150,000
5
0.353822= 23.5881
0.015
t
t
t
t
P e P
e
e
e
t
t
= =
=
=
æ ö÷ç =÷ç ÷çè ø
» 33 years 0 is 1990, so
the population will reach 150,000 in the year 2013.
t =