Chapter 2.2Algebraic Functions

Definition of Functions

A from to is from to where to each , therecorresponds

function

exactly one

a relation

such that

, .

fa

A B

bA

a

B

b

AB

f

Definition of Functions

no twoA func

ordertion is a se

ed pairs havt of ordered pairs in

whi e thesame first compo

ch nent.

Example 2.2.1

2

function

Identify if the following sets are functionsor 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

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

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

2

2 2

11. , 4 2

12. , 14 9

function

not a function

x y y x

y xx y

Notations

If is in a function then

we say that .

can be replaced ., ,

,

by

fx y

y f x

x y x f x

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

ff f

Vertical Line Test

A graph defines a function if eachvertical line in the rectangular coordinatesystem passes through at most one poi on the gr

ntaph.

Example 2.2.2Use the vertical line test to determineif each of the following graphs representsa function.1.

function

2.function

3.

not afunction

Algebraic Functions

can be obtained by a finite combinationof constants and variables together withthe four basic operations, exponentiation,or root extractions.

Transcendental Functions

those that are not algebraic

Polynomial Functions

11 1 0

General Form:

...

Domain:

If 0, the polynomial function issaid to be of degree .

n nn n

n

y f x a x a x a x a

a fn

Constant Functions

Form:

, where is a real number.

Graph: Horizontal Line

y f x C C

Dom f

Rng f C

Example 2.2.3

Find the domain and range then

sketch the graph of 3.

3

f x

Dom f

Rng f

Linear Functions

Form:

where and are real numbers, 0

Domain:Range:

Graph: Line

y f x mx b

m b m

Example 2.2.4

Find the domain and range then

sketch the graph of 3 4.f x x

Dom f

Rng f

x 0 -4/3y 4 0

Quadratic Functions 2

2

Form 1:

Graph is a parabola.0 : opening upward0 : opening downward

4Vertex: , or ,

2 4 2 2

y f x ax bx c

aa

b ac b b bf

a a a a

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

aDom f

ac bRng f y y a

a

ac by y a

a

Example 2.2.5

2

2

2

Find the domain and range then

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

2 4 2

vertex: 2,6 Axis of symmetry: 2

f x x x

x

x 1 3y 5 5

2

2

1 4 1 2 5

3 4 3 2 5

2x

6

Dom f

Rng f y y

Quadratic Functions

2Form 2:

vertex: ,

y f x a x h k

h k

Example 2.2.6

2

2

Find the domain and range then

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

22 1

vertex: 2, 1 Axis of symmetry: 2

f x x

x

x -3 -1y 0 0

2

2

3 2 1 0

1 2 1 0

2x

1

Dom f

Rng f y y

Maximum/Minimum Value 2

2

2

If ,

4vertex: ,

2 4

0 : The lowest point of the graph isthe vertex.

4 is the smallest value of .

4

f x ax bx c

b ac ba a

a

ac bf

a

Maximum/Minimum Value 2

2

2

If ,

4vertex: ,

2 4

0 : The highest point of the graph isthe vertex.

4 is the highest value of .

4

f x ax bx c

b ac ba a

a

ac bf

a

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

gx

Cubic Functions

3Form: y f x a x h k

Dom f R

Rng f R

x -1 0 1y -1 0 1

Example 2.2.8

3Consider

, 0,0

f x x

Dom f R

Rng f R

h k

x 1 2 3y 4 3 2

Example 2.2.9

3Consider 3 2

, 2,3

f x x

Dom f R

Rng f R

h k

Rational Functions

Form:

, are polynomials in degree of 0degree of 1

P xy f x

Q x

P Q xPQ

Rational Functions

The domain of a rational function isthe set of all real numbers except thosethat will make the denominator zero.

Example 2.2.10

2

Determine the domain of the followingfunctions.

11. 3

34

2. 222 2

2, 22

xf x Dom f

xx

g x Dom gxx x

g x x xx

2

2

13. 1, 1

1

even if1 1 1

, 11 1 1 1

xh x Dom h

x

x xh x x

x x x x

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

Example 2.2.11

Determine the equation of the vertical2 5

asymptote of .3 1

1 will make the denomiantor 0 so

31

the vertical asymptote is .3

xf x

x

x

Asymptotes

Consider the graph of

where and are polynomials

with degrees and , respectively.

P xf x

Q x

P x Q x

n m

Asymptotes

The of the graph is0 if

if

where and are the coefficients

of an

hor

d

izontal

.no horizontal asymptote if .

asymptote

n m

y n ma

y n mb

a b

x xn m

Example 2.2.12

2

2

Determine the equation of the horizontalasymptote for the following.

2 51.

3 14

2.21

3

23

no H.A

. 01

.

xf x

xx

g xxx

y

xyh x

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.

11.

2a. 2

b. V.A.: 2c. H.A.: 1d.

xf x

xDom f

xy

2x

1y x 3 4y 4 2.5

X 1 -1y -2 0

e. 1Rng f

2x

1y

2 2 242. 2, 2

2 2

a. 2

b. V.A.: nonec. H.A.: noned.

x xxg x x x

x x

Dom g

x 0 2y -2 0

2, 4

e. 4Rng g 2, 4

2

1 1 13. , 1

1 1 1 1

a. 1, 1

b. V.A.: 1c. H.A.: 0d.

x xh x x

x x x x

Dom h

xy

1x

0y x 0 1y 1 0.5

x -2 -3y -1 -0.5

1,0.5

1e. 0,

2Rng h

1x

0y 1,0.5

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

Square Root Functions

The domain of the square root function is theset of permissible values for x.

The expression inside the radical should be greater than or equal to zero.

| 0Dom f x P x

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

Example 2.2.15

7,4

3,2

4,3

3 2

3,

2,

f x x

Dom f

Rng f

x 3 4y 2 3

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

Example 2.2.17

2g 9

3,3

3,0

x x

Dom g

Rng g

x -3 0 3y 0 -3 0

3,0

0, 3

3,0

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

Conditional Functions

1

2

Form

condition 1condition 2

condition n

f xf x

f x

f x n

Example 2.2.18

3

2

2

3

Given that

5 if 51 if 4 2

3 if 2

find

1. 4 3 4 13

2. 0 0 1 1

3. 8 5 8 40

x xf x x x

x x

f

f

f

Example 2.2.19

For the following items,a. find the domainb. find the rangec. sketch the graph

3 2 if 11.

2 if 1x x

f xx

Dom f

x 0 -2/3y 2 0

1,5

5Rng f

2

2

1 if 02.

3 1 if 0

1 if 0

x xg x

x x

Dom g

y x x

Rng g

2

1 if 2

3. 4 if 2 21 if 2

2,2

, 1 0,2

x x

h x x xx x

Dom h

Rng h

Absolute Value Functions

Consider

if 0if 0

0,

y f x x

x xy f x x

x x

Dom f

Rng f

if 0if 0

x xy f x x

x x

0,

Dom f

Rng f

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

Example 2.2.20

Find the domain and range thensketch the graph of the given function.

1. 2 1

vertex: 2,1

1

f x x

Dom f

Rng f y y

x 0 4y 3 3

2. 2 3 7

3 7 2

73 2

37

vertex: ,23

2

g x x

x

x

Dom g

Rng g y y

x 0 3y -5 0