# Term

Rational Function
-

ta-
polynomial
-> functions
A rational function has w(x) P(x)
=

the following form: Q(X) -

P(x) and Q(x) does not have a common factor.

Let's checke the domain

x
=
0

f(x) =
4 -

d(x) =

x3 xts
The domain of a rational
function is made up of D : IR = O
all real numbers except D :
IR F-3 1R =3
those for which the g(x) x1z
=

denominator is zero.
D : IR F-2

And the simplest function

= f(x)
Domain :

IR Fo
=

4
Range : IRFO

x f(x) x f(x)
-0 1 .
-

10 0 1 . 10
-
0 .
01 +100 0 .
01 100

-
0 .
001 -
1000 0 001. 1000
f(x) ->
-

- when x- o

f(x) ->
+

- when x- of
y
fix
then
I

-----
↳
x- ot

f(x) -
0

when x -> 1

X
f(x) ->

↳
0

when x ->
-
-

f(x) - >
-

when x -> o
1

* Note (arrow notation)

x >
-
a X approximates to a by the left
t
x -
>
a X approximates to a by the right

x -

> - x X tends to - infinite

x - +

I X tends to + infinite
 Definition of a vertical
asymptote
The line x=a is a vertical asymptote of the function f(x) if
y approximates to +- infinite when x tends to be a

Definition of a horizontal
asymptote
The line y=b is a horizontal asymptote of the function
f(x) if y approximates to b when x tends +- infinite.

Example O

4 --
- Er

:
Horizontal 2x -

4x 5 +

.
=
z
t =

I
- x
2
-
2x +

1 1 -

E -
-
t
z
. . . . . . . . . . . .

r(x) = 2x -
4x +

5
=

2)
↳
x 2x +
1

Factors
vertical ((x
2

-
-

1)
x

Domain :

1R F I
Range
:
12 , %) /x> 2
 Example
#

E a Zab b2
*
+ +

r(x) = 3x2 -

2x -
1 (a +

b)(a +

b)
2x2 +

3x -

1) Vertical asymptote
( -

3 ,
-

2)u( -
z , /z)w(k 1) ,

We obtain the factor 2x +

3x -
2

of the denominator ↓
(2x -
11(X +

2 )
Vertical asymptote 2x -
1 =

0 x
+

2 =
0

2x 1 x 2
Yz
-

=
=

x
=

x
=

12
X 2
-

Yz
=

D =

1R = = -

2) Horizontal asymptote
3x2 2x 1
the
-
-

2
we use main
2x2 +
3x -

coefficient of te
3x-2x-1 . z
numerator and denominator
-

2x2 +
3x -

242
-
2 Yz 3 -

-
2 +

2-
3/2
- -
o
O I
2
3 -
* ↳
2
2 +
-
>
-
R
=

1R F 4z
D = 1R F -
2 ,
El
 More about horizontal asymptotes

n 1

x
-

~(x) =

an
+

AnX + . . . + a , + Go

bm x +

bm *** + . . .
+

b, x +

bo

a) If n <m , so the horizontal asym .

Y = 0

b if n
=

m , so the horizontal asym y =

E
c) if n> m
,
so the horizontal asym
does not exist

x =
0

a) f(x) =

x4 -
. . . . . . .

b) f(x)
2 E
=
=

f(x)
.

a = = E X
5
= 5
31
4 X
7

x5
I
2
x

5
- -
=

5
=

h
#
#
 How to graph ?
1. Factor: get the factor of the numerator and denominator.
2. Intersection points.
3.
4.
Vertical asymptote.
->
Horizontal asymptote. X- intersection
5. Sketch the graph.
y
- intersection

But... first let's check a

graph

-
Vertical

I
2

4
y -
intersection

·
L
Horizontal

x -
interses
 The first graph
r(x) = 2x +

7x -

4
x
+
x
-

1) Obtain the factors

2x 7x 4 x 2
x
+ -
-
+

(2x -

1)) x +

4) (x -

1)(x 2) +

2) Intersection points

y = (2x -

1)(x +

4)
(x -

1)(x +

2)

x -

intersection yintersection
-

/Zeros numerator) ·
(substitute x with of
I #4
2
2x -
- I
4)
+

(2x 1)(x
-
+

2
-
2
* ⑭
-

2x -
1 = 0 x +

4 = 0
y
=

2
2x =
1 x
-

=
4
x
=

1/2

3) Vertical asymptotes
*

Where the denominator is not defined -

ji
! Ii
X -
1 =
0 x
+
2 =

X =
1 x
=

= 2

x
-
> a - > f(x) =
x/ -
0

-
x - D
3 .
1) Behavior near Vertical asymptotes
(2x 1)(x 4) +

f(x)
-

* -> to I near
(x 1)(x 2) -
+

the value of the (2(0 9) 1) (0 9 4)

.
- . +

-
(t(H =( )
-

( 1 )
10 1) (0 9 2)
asymptote 9
+
-

.
- .

x -
> 1 -
0 .
9 -

f(x) - -

f(x)
+

x -
1 -
( . 1 +
-
+

x -

> z- -

f(x) - -

I
x -

> -
2
+

f(x) -

> 0

4) Horizontal asymptote
r(x) = 2x +

7x -
4 -
I
= =
2
z
-

2 2
x x
-
+

We can evaluate some

5) Sleetch the
*

graph points
I

B
I
I

h
f(x)
I

X
I I

6 93 -
-

0 .
I
-en

-
-
-

!"
-
3 -
1 .
75
-
I 4 3 .

1 6 24
5

. .

2 4 3 .
D : =- ,,

3 3 .
5 R :
IR
 The second graph
r(x) =

5x +

21

x2 +

10x +

25

Obtain the factors

r(x) =

5x +

21
(x +

5)

2) Intersection points

X
=

5x

(x
+

+
21

5)2
+ 25

x -

intersection y
-
intersection
5x +
21 = 0
-z +
=
2
5x =
-

21
25
o+ 10(0) 25
245
+

x -

3) Vertical asymptote
x 5 0
+

x
-

=
5
!
 .
3 1) Behavior near vertical asymptotes
-

-+- --

x -
> 5 -> -
5 .

1 2
( +)
( -
5 1
-
+

5)
f(x) -> -
I
4 y
+

6
t
-

-
I
-

S
T
x - > -

5 ->
-
4 .

f(x) -> -

00

4) Horizontal asymptote
f(x) =

5x +
21 y =
0
2

x
+
10X +

25

5) Sketch

--
x f(x)
-
15 -

0 .
5
- 10 -1 . 2

-
3 1 .
5
-
I 1.
O

3 0 . 6
D :
IR F -

5
5 0 . 5

L 0 0 3 .

R : (-g ,
1 .

55
 ++
Example

r(x) = x2 -
3x -

2x +

4x

1) Obtain the factors

r(x) =
⑰x1) +

2x(X +
2)

2) Intersection points
y = (x 1)(X 4)
+
-
3x-4 4X

2x)x 2) +

X -
intersection y - intersection
x
+
1 0 x -

4 =
0 2
4
=

-4
-
-

X =
-

1 x = 4 =
210) +
4(0)
undet

3) Vertical asymptote
2x = 0 x +
2 = 0

x
=
02 x
=
-
2

x
=
0
3 .

1) Behavior near vertical asymptotes

(x 1)(x
4)
+
- -

x -
> - - 2 1 .
f(x) - x +

0 2x(x 2)+

t
x -> -
2 -x -
1 9
.
f(x) -> -

-
=

= = -

x -
0 =
- 0 .
1f(x) - > +
0

+
x -
> 0 0 . 1 f(x) - -

4) Horizontal asymptote
0

Yz + I

-I
5) Sketch 1
⑧

-
X

2
3
.
5
f(x)
2

3
.

33
9 -

-Il -
I

+-----

A iS
.

⑧ ⑧

3
2
-
I
5 1 .
-

3
-

-0 .

I -
I

3 -0 13
!
.

5 0 04 .

D IR : = -2
,
0
-

B 1 = Y2
:
 Removable discontinuity
It’s a point that is not defined in the rational function.

w(x) = x +
x -
z

x +
3x -

1) Obtain the factors

r(x) I
=
/
/
***
=
3)
2) Intersection points ,

x
=
(xf1)(x 2) +

(x
/ 1)(X 4)
-
+

X -
intersection y intersection
== =
X + 2 = 0

x
=

=
2

3) Vertical asymptotes

(x+1)(x 2) +

x =

(x+ 1)(x 4) x 4 0
+
+
=

x 4
=

-
.
3 1) Behavior near vertical
asymptotes
x -

> -
4 - x -

4 . I #2 = =
+

f(x)
-
4 1
. + 4
-
> y

t
↓ 4 4
4 !
- - -> -

3 .
= =
-

f(x) - -
0

4) Horizontal Asymptote
y = x +

x -
2 1 =
1
I
x +

3x -

5) Stretch graph
I
+

i
--------
. . . . . . .

" I
mos
I

!
v -
D
 And or the diagonal asymptote
If r(x)=P(x)/Q(x) is a rational function in which the degree of the
numerator is one more than the degree of the denominator, we can
use the Division Algorithm to express the function in the form:

R(x)
r(x) ax b
s
+
+
=

Q(x)

* i coefficient is smaller than the Q(x)

* FO
B(x)
* x -
> D
-
-> O
Q(x)
SO . . .

* x
=

r(x) -

y =
ax +

Diagonal asymptote or oblique asymptote

Division algorithm

r(x) =
x -

4x -
5

x 3
-

X -
1

(x 3)-

x -

4x -

3
2
x
-

3x

x 5
-

- x +

⑧
1) Obtain the factors x
#
5
x -
3

5)
Y
-

2) Intersection points

y
=

(x +
1)(X -

5)
(x -

3)

x -
intersection y - intersection
x +
1 =
0 x -
5 = 0

5
5
-

-
1
-
x x 5
- =

3) Vertical asymptote
)
x -
3
=

0 ⑮x
3
y =

f(x)
-

x -

> 3 -
+
0

x - 3t f(x) -
> -
0
4) Horizontal asymptote

Under

5) Diagonal asymptote
⑧
r(x) =
x -
1 -

x
-

x 1
y
-
=

6) Sketch

rig
x f(x)
-

2
-

1 .

!
5

i
-

2 .
33
 The form of rational functions

f(x) =
a
x -

x in
f(x)
-

)
a(x
-
-
=

Verasy) (verAsyl

f(x)
by(x
=

(x -
-

c)

f(x) =

a(X -

b)
(x -

C)(X -

d)

f(x) =

a(x b)(x c)
-
-

(X - d)(x -
e)

The most basic form

f(x)
1
=
 ⑦ f(x) b move->
-

,xqb)
=

+ b move -

f(x) =
2

X 3

f(x)
B
=

-
3

f(x)
Y
=

f(x) =
2

x +
3

f(x)
Y
=

+
3

f(x) =

x4 +

3
⑪ f(x) =

!
qx Y ,
b)

f(x) = =(x 3) -

(x +

4)

f(x) =
3(x -

3)
(x +

4)

P(x , , y)

y
a
=

x
=
 Modular 4

E xponential functions
Definition
-
a)0
f(x)
x

f(x) =

a
*
a =
1
~
=

↓
↓
x
=

1 Constant function
base
all have
*

Similar
Graph of Exponential functions characteristics
*

y a
=

=Iy
2 0 b 2
4X
ii
-

q(x) I
=
-
2 2
0 =

1
4 4 44
=

-
4 4 16
=
=
=

f(x)
Y
82 42 !20
-

2
=

2 2 =
=

=
12 = zz =
4

TY
-
*

=
A
 x
y a
=

-
2 -

I 0 l 2 3

=
-
X -- 2
-

g(x) 4 4 4 4
=
=

-
f(x) =

2
x
2 2 42 zzEi =
= 1
2

2
=1
4

-
2
Now Domain etc
range
..

, ,
.

Domain : IR

Range : (0 1),

a -

0 as x - -

f(x) ↳
-

Domain :

IR

Range :
20 ,
+

1)
Y
a
y 0 asx
=

y
-
+
-

↳
 Transformations of graphs

Horizontal shift ->

t

b
f(x)
x +

+
b left
-

b
right

Vertical shift
f(x) =
a -

b down

+b up

Reflection in x-axis

f(x) =

-
a

-
a

↳ X-axis
reflection
I
↓
-
22
4
 Reflection in
y-axis

f(x) =

a
Y

=
y-axis
reflection

Logarithmic Functions 101

Logarithmicactions
The logarithmic functions the
wit
are the inverse functions of
the exponential functions.

#
X > 0
X - I X
T
ofa
Lx
1

-
7

a>0
I
a Fl X = a

Examples
1) f(x) 3232 2
=

log2 x x = =

↓ X
2) f(x) =

log x =
o 100
= 10
,

f(x)
3) f(x) =

log4 x x
=

2 2
=

4
 The natural base C

f(x) f(x)
*
=
a
=
ex
In some cases the
most convenient 2 =
2 .
718281828 . . .

choice for a base is

the irrational number
e ↓
natural base

The natural exponential function

x>0

y
=

In c/ y =

loge X

↓
↑
↳↑X L

X = e

i
 But what are

logarithmic functions ?
Example I
Example #I

dB Richter Scale

x(dB)
10log(i) log(E)
=

R =

Volume
absolute relative (dB) 3 .
6

1000 30 is 10X
bigger
100 20 than
10 10 2 .
6

I O

0 .
1 -

10

0 . 01 -
20

0 .
001
-

30

Example

pH of a
liquid
Hydrogen ions

pH =

-log [H +
I
+ = 1
Logarithmic Functions 102

M
3) 109
,
+

215 Basic properties

The same for
0

1) log 1 0 a 1
=

- =

In
a

2) log ,
a =

1 -> a =
a

a'08a
*
3)
log ,
a =
x and = X

4) If log ,
x
=

log, Y ,
then x
=

Change-of-base formula
a F
I b = 1

Base b Base 10 Base e

log, X= logpx log,* =

logx logx= Inx

In a
log , a
log a

log , 25 =
log ,
25
log 12 =

log
- ,

109 , 0
4 log 2

log , 25 In 1112
25
log , -
=
-

In 4 in 2
 + Properties
a be a positive number n -> real number

a #
I
U ,
e ->
positive real

numbers

Base
A

a Natural Logarithm
Product
Property log (UM) =

log +loger
,
Invert =

In the

In (E) =
Quotient
log(E) log-loger = Inu-her
Property
loge
Inc
Dower
logu nloga ninv
=

Property
Example

a) (n(6)
In (2 3) .
bl In()
In (2) -

In (27)
In 2 +
In 3

4
log ,
3
=

log
34(3(n x +

in(x2 +

3))
 Examples the revenge
+

3 points if someone
gives me the exact result
with complete process

al log , (5) b) In e -

In e
I
5(3) ) 6 In- zie
log
: log,

' logg's 6 -
z =
⑰

⑬ In
loge
=

->
expand
5/n (x +

2)

In (x+ 2) condense
 Rewriting Logarithmic Expressions

The properties of logarithms are useful for rewriting logarithmic expressions in forms
that simplify the operations of algebra. This is true because these properties convert
complicated products, quotients, and exponential forms into simpler, sums,
differences, and products, respectively.

a) logy 5x y b) In <- 5-5

7

lg
5 +
log
,
+log "
,
In
(35)
In 13 x -

5) -

In 7

In (3x-5) -
In 7
 Condensing Logarithmic Expressions

altlog + blog?x1)
b) zIu(x+ e

In (x 2) +

-
Inx

log(x )(x + 1) In (x
-
+
2)
X

log(x(x + 1)3

d [log x +

log,(x 1)]
+

: [logx(x+ 1)]

log(x(x 111 +

logix

⑤ log ,
(x(x+1) ->
log, x
 Aleks

3 -= x =

I
*
210 = 100

1092 log
36
36-2log log
I
2 -
=
4 3

4 ,
, ,

o
5
10924 X
- =

2
Y
=
4 x
=
L
log 36-log
,
,3 log
,
4 =

-o 32 +
-
-

8 3
+
= -
5
log ,
36-log ,
9 2
=

2 =

4 .
2 in 9 he +
8 +
9 =
17

5
log. =
I
IX
*
=
= -

2
2

7
log
!
It
t
1 .
209

x +
1

(0g4 19 =

15

⑧ (x+
1) log, 19 =
15

(x
1)() 15
=

X 6 062
=

x 1)
=
+

=
x
 Review
4

1)
zi""us
X

1082

log2 x +

log(y +

2) -

[blog (z 2)+ +

log(x 4)]
+

2
,
108
"
zlog , (2 +

10) +
log ,
X -

[4log(y -
3) +

log(z 14)]+

3) In 2) /x-27 3)

(w 23
+

3(n(x 2)+ +

z(n(y -

2) + (n(z 3) -
-

[3(n(z +

2) +

(na]

16x(x +
2)
=

Xog(x +

6)

x2 +

2x =

x +

6
 Exponential and
logarithmic equations
One-to-one Inverse
properties properties

loga
·

* *

a =
aY only x =

Y =X

log-logx only log

*
x
x Y
=

a =

① Solving simple equations In= loge

X 5
X
2 = 32 -> 2 =

2 -> x
=

Inx -

1n3 =

0 -
> (nx =

(n3 -
x
=

(i) X 2
9
-
=
--
3 I
3 -> x
=

- 2
*
e
=

7 -

Ne = In7 -> x
=

In7
3
/Inx
3 -

In x
-

3 ->
=

= ->
x
-

e C =
e

logx
logx - 1
=

- > 10
=

10
->

x
=
10
x
=

Strategies for Solving Exponential Equations and Logarithmic Equations

1. Rewrite the original equation in a form that allows the use of the One-to-One Properties
of exponential or logarithmic functions.
2. Rewrite an exponential equation in logarithmic form and apply the Inverse Property of
logarithmic functions.
3. Rewrite a logarithmic equation in exponential form and apply the Inverse Property of
exponential functions.
 ② Solving exponential equations
* -
3x -
4 -x
2

a) =
e In
/
/
e b) 3(2Y) =

42

Inex
- - In/
/ e
-

3x -

4
2
x
=

44y
x4
-

2 Y
-
X
=

-
3x -
4 2
=

14
2

x - 3x -
4 =
0
109/
,
* =

log2
(x 1)(X 4) 0
=
-
+

x
=
- 1
x
=

log2 14 = 14

x
=

4 log 2

③ Solving an exponential equation

b) z(3t 5)
-

a) e
+

5
=

60 -

4 =

11
55
31
+
e =
60 -
5- 23 15

- -
3
Ix =

In 55 3
=

I
=
x In 55
log log ,
=

t- 5 =

log ,
t =

log , 7 .
5 +

5
-
! 5
+

=
⑭ Solving an exponential equation of quadratic type
2x
Solve c - 3e +
2 =
0

In e
1 -
3x + Inz =
so
Ino

2x -

3x +
(nz = 0

x +
1nz =
0

log2 X
-
5

x
=
In 2 -2 - 2

25
-
5

X
=

x 2
=

Obtain the x values for

which y = 0

x
=

x =
0 . 693
 - X

500e = 300

-->
304/54
=

Inc
/ =

In 3/5 -> In
/ =

In
5

In 15 *
=

In
* =

3/5

I =

In x
=

In
-
6 4X

2x
-

50x +

39
64 =

(26)6
ax

2x2
-

50x +

39
=

2
36 -
34X * -50x +

34
2 - 2
2

36 -
34X x -
50x +

39
I
oy
-
1092
// 2

36 -

54x =

x2 -
50x +

39
*

x
+
4x +
3

(x +

3)/x +

1)

x
=

- 3
x
=

- 1
x
= 0

5 lite)
20
=

↓
25

250
I =

10
N2 *
100 -
e -
25 In e =
In 75
T
75
-

N2 -> I =
In 75
-

e
=

25 -

100
x x
=

2 (n75
575
=

- -
C

exe I 75
2t

11t q4) Wi

12 0
=
2

~2t
I

"
=2

2
10g)2t =
t

=
 Solving Logarithmic Equations
* you can write in exponential form
in X 3 3
In x
=

3 - e =
e -> X = e

①
Solving equations
a) Inx = 2 3) log (5x 1) -
=

log (x 7) +

, ,

IInx 2

log , (5x 1)
logs (x 7)
-S
C
C /
= -
+

2 3 -
/
3
x 2
=

5x -
1 =

x + 7

5x -
x
=

7 +
1

4x
=

8x =
5 = 2

C log ,
(3 x +

14) -

log, 5 =

log , X

log( ** ) =
log, x

** )
glog( =

60
-
3) 4 =
2x
5

14
x
=
=
3x 10x
=

14 =

7X
② Solving equations
21n b)
a) 5 x 4 <log 3x 4
= =

21nx =
4 -

5 109 , 3x =

E
21nx
91095
= -

1 3x 2

5
=

Inx = 3 x
=

25

42 x
=
25
/3
x
=

③
Checking for Extraneous Solutions
Solve log5x +
log(X-1) 2
=
:

log 3x(x -
1) =

2 5x2 - 3x -100 =
0

log 5x -
5x =

2 Y -
x
-
20 =

2 (X -

3)(x +

4) =

5x2 -
5x =

10
x 5
=

5x2 -

5x = 100
x
=

-
4
 Exponential and
logarithmic models

five most common models

& Exponential growth

model
bx
y = ae b>0

② Exponential decay
model

y
=
ab b>0

③ Gaussian model
- (x -

b)Yc
ac
y =
⑭ Logistic growth
model

a
y =

1 +
be

⑤ Logarithmic models

y =
a
+

b(nx

y =
a +

blog x
 Exponential growth and decay model

Estimates the number (in millions) of U.S. households with digital television from 2003 through
2007 are shown in the table. The scatter plot of the data is also shown:

An exponential growth model that

approximates these data is given by:

0 1171 t
D
.

= 30 .
92 e
31 + 77 -

Where D is the number of households (in millions) and t = 3 represents 2003. Compare the
values given by the model with the estimates shown in the table. According with this model,
when will the number of U.S. households with digital television reach 100 million?

Solution

let D =

100 and solve for t

 Exponential growth
In a research experiment, a population of fruit flies is increasing according to the law of
exponential growth. After 2 days there are 100 flies, and after 4 days there are 300 flies. How
many flies will there be after 5 days?

Solution
Let y be the number of flies at time t. From the given information, you know that y = 100
when t = 2 and y = 300 when t = 4. Substituting this information into the exponential
growth model produces:
 Carbon Dating

In living organic material, the ratio of the number of radioactive carbon isotopes (carbon 14) to
the number of nonradioactive carbon isotopes (carbon 12) is about 1 to the 10**12. When
organic material dies, its carbon 12 content remains fixed, whereas its radioactive carbon 14
begins to decay with a half-life of about 5700 years. To estimate the age of dead organic
material, scientists use the following formula, which denotes the ratio of carbon 14 to car

R =

48223

Estimate the age of a newly discovered fossil in which the ratio of carbon 14 to carbon 12 is:

R =

3
 Exponential decay
The value of b in the exponential decay model determines the decay of radioactive isotopes. For
instance, to find how much of an initial 10 grams of 226Ra isotope with a half-life of 1599 years
is left after 500 years, substitute this information into the exponential decay model.

Gaussian Models
This type of model is commonly used in probability and statistics to represent populations that
are normally distributed. The graph of a Gaussian Model is called a bell-shaped curve. For
standard normal distributions, the model takes the form of:

e
y x
=

The average value for a population can be found from the bell-shaped curve by observing
where the maximum y-value of the function occurs. The X-value corresponding to the maximum
y-value of the function represents the average value of the independent variable-in this case, X.
 Gaussian Model
In 2004, the Scholastic Aptitude Test (SAT) math scores for college-bound seniors roughly
followed the normal distribution given by:

y
=

0 .
00351X-5184/25 ,
942
2001 x1800
Where x is the SAT score for mathematics. Sketch the graph of this function. From the graph,
estimate the average SAT score.
 Logistic Growth Models

Some populations initially have rapid growth, followed by a declining rate of growth, as
indicated in the graph. One model for describing this type of growth pattern is the logistic curve
given the function:

Where y is the population size and x is the time. An example is a bacteria culture that is initially
allowed to growth under ideal conditions, and then under less favorable conditions that inhibit
growth. A logistic growth curve is also called a sigmoidal curve.
On a college campus of 500 students, one student return from a vacation with a contagious and
long-lasting flu virus. The spread of the virus is modeled by:

Where y is the total number of students infected after t days. The college will cancel classes
when 40% or more of the students are infected.
A) How many students are infected after 5 days?
B) After how many days will the college cancel classes?
 Logarithmic Models

On a Richter scale, the magnitude R of an earthquake of intensity I is given by

R =
I

Io

Where Io = 1 is the minimum intensity used for comparison. Find the intensities per unit of area
for each earthquake. (Intensity is a measure of the wave energy of an earthquake).

a. Northern Sumatra in 2004: R=90

b. Southeastern Alaska in 2004: R=6.8