2.2 algebraic functions

Chapter 2.2 Algebraic Functions

1

Definition of Functions

A from to is a relation

from to where to each , there

correspo

function

exactly nds such that

, .

one

f

a A

b

A B

A B

a

B

b f

2

Definition of Functions

no two

A func

order

tion is a se

ed pairs hav

t of ordered pairs in

whi e the

same first compo

ch

nent.

3

Example 2.2.1

2

function

Identify if the following sets are functions

or not.

1. 1,3 , 2,5 , 3,8 , 4,10

2 not a funct. 1,1 , 1, 1 , 2,2 , 2, 2

3. , 2 5

ion

function

function4. ,

x y y x

x y y x

4

25. , 5

1,2 and 1, 2 are

both in the relation

6. , 5 1

7. , 6

0,6 and 0, 6 are both

in

not a function

function

not a f

the relat

unct

n

on

i

i

o

x y x y

x y y x

x y x y

5

8. , 3

0,0 and 0, 1 are both

in the relation

9. , 5

5,1 and 5,2 ar

no

e

t a function

not a function

functi

both

in the relation

10. , on

x y y x

x y x

x y x y

6

2

2 2

11. , 4 2

12. , 14 9

function

not a function

x y y x

y xx y

7

Notations

If is in a function, say then

we say that .

can be replaced by,

,

, .

fx y

y f x

x y x f x

8

Notations

2

2

2

2

Given , 3 1

3 1

3 1

2 3 2 1 13

2,13 2, 2

f x y y x

y x

f x x

f

f f f

9

Algebraic Functions

can be obtained by a finite combination

of constants and variables together with

the four basic operations, exponentiation,

or root extractions.

10

Transcendental Functions

those that are not algebraic

11

Domain and Range

The domain is the set of all values of the independent variable

permissible

resulting

.

The range is the set of all values of the dependent variable.

Example 2.1.5

Identify the domain and range of the

following functions.

1. , 2 1S x y y x

Dom S

Rng S

2

2

2. ,

0,

3. , 4

4,

T x y y x

Dom T

Rng T

U x y y x

Dom U

Rng U

24. ,

1

1

0

5. , 1

1 0 1,

1 0,

V x y yx

Dom V

Rng V

W x y y x

x Dom W

x Rng W

26. ,

0,

0

7. , 2 3

0,

X x y x y

y x Dom X

x Rng X

Y x y y x

Dom Y

Rng Y

8. , 5 4

4,

9. , 5

0

0,

,5

Z x y y x

Dom Z

Rng Z

A x y y x

x

Dom A

Rng A

Polynomial Functions

11 1 0

General Form:

...

Domain:

If 0, the polynomial function is

said to be of degree .

n nn n

n

y f x a x a x a x a

a f

n

18

Constant Functions

Form:

, where is a real number.

Graph: Horizontal Line

y f x C C

Dom f

Rng f C

19

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y

Example 2.2.3

Find the domain and range then

sketch the graph of 3.

3

f x

Dom f

Rng f

20

Linear Functions

Form:

where and are real numbers, 0

Domain:

Range:

Graph: Line

y f x mx b

m b m

21

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y

Example 2.2.4


sketch the graph of 3 4.f x x

Dom f

Rng f

x 0 -4/3

y 4 0 22

Quadratic Functions

2

2

Form 1:

Graph is a parabola.

0 : opening upward

0 : opening downward

4Vertex: , or ,

2 4 2 2

y f x ax bx c

a

a

b ac b b bf

a a a a

23

Quadratic Functions

2

2

2

Form 1:

Symmetric with respect to: 2

axis of symmetry

4 if 0

4

4 if 0

4

y f x ax bx c

bx

a

Dom f

ac bRng f y y a

a

ac by y a

a24

Example 2.2.5

2

2

2


sketch the graph of 2 4

4 2 1, 4, 2

4 1 2 44vertex: , 2,6

2 1 4 1

6

Axis of symmetry: 2

f x x x

f x x x a b c

Dom f

Rng f y y

x

25

-4 -3 -2 -1 1 2 3 4

-3

-2

-1

1

2

3

4

5

6

7

x

y

2 4 2

vertex: 2,6 Axis of symmetry: 2

f x x x

x

x 1 3

y 5 5

2

2

1 4 1 2 5

3 4 3 2 5

2x

6

Dom f

Rng f y y

26

Quadratic Functions

2Form 2:

vertex: ,

y f x a x h k

h k

27

Example 2.2.6

2

2


sketch the graph of 2 1

2 1

vertex: 2, 1

1

: 2

f x x

f x x

Dom f

Rng f y y

AOS x

28

-4 -3 -2 -1 1 2 3 4

-3

-2

-1

1

2

3

4

5

6

7

x

y

22 1

vertex: 2, 1 Axis of symmetry: 2

f x x

x

x -3 -1

y 0 0

2

2

3 2 1 0

1 2 1 0

2x

1

Dom f

Rng f y y

29

Maximum/Minimum Value

2

2

2

If ,

4vertex: ,

2 4

0 : The lowest point of the graph is

the vertex.

4 is the smallest value of .

4

f x ax bx c

b ac b

a a

a

ac bf

a

30

Maximum/Minimum Value

2

2

2

If ,

4vertex: ,

2 4

0 : The highest point of the graph is

the vertex.

4 is the highest value of .

4

f x ax bx c

b ac b

a a

a

ac bf

a

31

Example 2.2.7

2If 1 10 find the maximum/

minimum value of .

vertex: 1,10 0

the maximum value of is 10.

the maximum value is obtained when 1.

g x x

g

a

g

x

32

Vertical Line Test

A graph defines a function if each

vertical line in the rectangular coordinate

system passes through at most one poi

on the gr

nt

aph.

33

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y

Example 2.2.2 Use the vertical line test to determine

if each of the following graphs representsa function.1.

function

34

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y2.function

35

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y3.

not a

function

36

Cubic Functions

3Form: y f x a x h k

Dom f R

Rng f R

37

x -1 0 1

y -1 0 1

-4 -3 -2 -1 1 2 3 4

-3

-2

-1

1

2

3

4

5

6

7

x

yExample 2.2.8

3Consider

, 0,0

f x x

Dom f R

Rng f R

h k

38

x 1 2 3

y 4 3 2 -4 -3 -2 -1 1 2 3 4

-3

-2

-1

1

2

3

4

5

6

7

x

y

Example 2.2.9

3Consider 3 2

, 2,3

f x x

Dom f R

Rng f R

h k

39

Rational Functions

Form:

, are polynomials in

degree of 0

degree of 1

P xy f x

Q x

P Q x

P

Q

40

Rational Functions

The domain of a rational function is

the set of all real numbers except those

that will make the denominator zero.

41

Example 2.2.10

2

Determine the domain of the following

functions.

11. 3

3

42. 2

2

2 22, 2

2

xf x Dom f

x

xg x Dom g

x

x xg x x x

x

42

2

2

13. 1, 1

1

even if

1 1 1, 1

1 1 1 1

xh x Dom h

x

x xh x x

x x x x

43

Asymptotes

The graph of

where and have no common

factors has the line verti a cal

asymptot if . e 0

P xf x

Q x

P x Q x

x a

Q a

44

Example 2.2.11

Determine the equation of the vertical

2 5asymptote of .

3 1

1 will make the denomiantor 0 so

31

the vertical asymptote is .3

xf x

x

x

45

Asymptotes

Consider the graph of

where and are polynomials

with degrees and , respectively.

P xf x

Q x

P x Q x

n m

46

Asymptotes

The of the graph is

0 if

if

where and are the coefficients

of an

hor

d

izontal

.

no horizontal asymptote if .

asymptote

n m

y n m

ay n m

b

a b

x x

n m

47

Example 2.2.12

2

2

Determine the equation of the horizontal

asymptote for the following.

2 51.

3 1

42.

21

3

2

3

no H.A

. 01

.

xf x

x

xg x

xx

y

xyh x

48

Example 2.2.13

For each of the following,

a. Find the domain.

b. Find the V.A.

c. Find the H.A.

d. Sketch the graph.

e. Find the range.49

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

11.

2

a. 2

b. V.A.: 2

c. H.A.: 1

d.

xf x

x

Dom f

x

y

2x

1y x 3 4

y 4 2.5

X 1 -1

y -2 0

50

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

e. 1Rng f

2x

1y

51

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

2 2 242. 2, 2

2 2

a. 2

b. V.A.: none

c. H.A.: none

d.

x xxg x x x

x x

Dom g

x 0 2

y -2 0

2, 4

52

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

e. 4Rng g 2, 4

53

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

2

1 1 13. , 1

1 1 1 1

a. 1, 1

b. V.A.: 1

c. H.A.: 0

d.

x xh x x

x x x x

Dom h

x

y

1x

0y x 0 1

y 1 0.5

x -2 -3

y -1 -0.5

1,0.5

54

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

1

e. 0,2

Rng h

1x

0y 1,0.5

55

Square Root Functions

We will consider square root functions that

are of the form

where is either linear or quadratic and

0 , .

f x a P x k

P x

a k R

56


The domain of the square root function is the

set of permissible values for x.

The expression inside the radical should be

greater than or equal to zero.

| 0Dom f x P x

57


Form 1:

Domain: 0

Range: if 0

if 0

Graph:upper / lower semi parabola

opening to the right or left

y a mx b k

x mx b

y y k a

y y k a

58

Example 2.2.14

Consider the function 3 2

| 3 0 | 3 3,

Note that 3 0.

Therefore 3 2 2

2,

f x x

Dom f x x x x

y x

y x

Rng f

59

Example 2.2.15

7,4

3,2

4,3

3 2

3,

2,

f x x

Dom f

Rng f

x 3 4

y 2 3

60


2 2

2 2

2 2

Form 2: , 0

Domain: ,

Range: 0, if

,0 if

y a x a

a a

a y a x

a y a x

61


2 2

2 2

2 2

Form 2: , 0

Graph: if

upper semi-circle

with center 0,0 and radius

if ,

lower semi-circle

with center 0,0 and radius

y a x a

y a x

a

y a x

a62

Example 2.2.16

2

2

2

2

Consider the function g 9

|9 0

| 3 3 0 3,3

Note that 0 9 3.

Therefore -3 - 9 0

3,0

x x

Dom g x x

x x x

x

x

Rng g

63


2 2

2 2

2 2

Form 3: , 0

Domain: , ,

Range: 0, if

,0 if

y x a a

a a

y x a

y x a

64


2 2

2 2

2 2

Form 3: , 0

Graph: if

upper semi-ellipse

with x-intercepts ,0 and ,0

if ,

lower semi-ellipse

with x-intercepts ,0 and ,0

y x a a

y a x

a a

y a x

a a65

Example 2.2.17

2g 9

3,3

3,0

x x

Dom g

Rng g

x -3 0 3

y 0 -3 0

3,0

0, 3

3,0

66

Conditional Functions

1

2

Form

condition 1

condition 2

condition n

f x

f xf x

f x n

67

Example 2.2.18

3

2

2

3

Given that

5 if 5

1 if 4 2

3 if 2

find

1. 4 3 4 13

2. 0 0 1 1

3. 8 5 8 40

x x

f x x x

x x

f

f

f

68

Example 2.2.19

For the following items,

a. find the domain

b. find the range

c. sketch the graph

69

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

3 2 if 11.

2 if 1

x xf x

x

Dom f

x 0 -2/3

y 2 0

1,5

5Rng f

70

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

2

2

1 if 02.

3 1 if 0

1 if 0

x xg x

x x

Dom g

y x x

Rng g

71

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

2

1 if 2

3. 4 if 2 2

1 if 2

2,2

, 1 0,2

x x

h x x x

x x

Dom h

Rng h

72

Absolute Value Functions

Consider

if 0

if 0

0,

y f x x

x xy f x x

x x

Dom f

Rng f

73

if 0

if 0

x xy f x x

x x

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

0,

Dom f

Rng f

74

Absolute Value Functions

Form:

Vertex: ,

if 0

if 0

y f x a x h k

h k

Dom f

Rng f y y k a

y y k a

75

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

Example 2.2.20


sketch the graph of the given function.

1. 2 1

vertex: 2,1

1

f x x

Dom f

Rng f y y

x 0 4

y 3 3 76

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

6

x

y

2. 2 3 7

3 7 2

73 2

3

7vertex: ,2

3

2

g x x

x

x

Dom g

Rng g y y

x 0 3

y -5 0 77

Challenge!

2

2

upper semi-circle

Identify the graph of the following functions.

1. 4

2 parabola

horizontal line

semi-parabola

li

. 1 2

3. 3

4. 1 2

15.

3ne

f x x

g x x

h x

j x x

xk x

Education

2.2 algebraic functions