2.2 algebraic functions
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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
Find the domain and range then
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
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
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
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
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
Square Root Functions
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
Square Root Functions
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
Square Root Functions
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
Square Root Functions
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
Square Root Functions
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
Square Root Functions
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
Find the domain and range then
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
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