AP CALCULUS AB

Chapter 1:Prerequisites for Calculus

Section 1.5:Functions and Logarithms

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

Finding Inverses

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 =

Section 1.5 – Functions and Logarithms

You try: Solve for x:1.

2. 3

ln 6 4

100x

x t

e

Section 1.5 – Functions and Logarithms You try: Sulmaz invests $18000 in an

account that earns 6.75% interest compounded annually. How long will it take the account to reach $4320?