chapter 2.2 algebraic functions. definition of functions
TRANSCRIPT
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Chapter 2.2Algebraic Functions
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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
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Definition of Functions
no twoA func
ordertion is a se
ed pairs havt of ordered pairs in
whi e thesame first compo
ch nent.
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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
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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
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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
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2
2 2
11. , 4 2
12. , 14 9
function
not a function
x y y x
y xx y
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Notations
If is in a function then
we say that .
can be replaced ., ,
,
by
fx y
y f x
x y x f x
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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
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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.
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Example 2.2.2Use the vertical line test to determineif each of the following graphs representsa function.1.
function
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2.function
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3.
not afunction
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Algebraic Functions
can be obtained by a finite combinationof constants and variables together withthe four basic operations, exponentiation,or root extractions.
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Transcendental Functions
those that are not algebraic
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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
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Constant Functions
Form:
, where is a real number.
Graph: Horizontal Line
y f x C C
Dom f
Rng f C
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Example 2.2.3
Find the domain and range then
sketch the graph of 3.
3
f x
Dom f
Rng f
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Linear Functions
Form:
where and are real numbers, 0
Domain:Range:
Graph: Line
y f x mx b
m b m
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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
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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
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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
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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
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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
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Quadratic Functions
2Form 2:
vertex: ,
y f x a x h k
h k
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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
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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
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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
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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
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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
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Cubic Functions
3Form: y f x a x h k
Dom f R
Rng f R
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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
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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
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Rational Functions
Form:
, are polynomials in degree of 0degree of 1
P xy f x
Q x
P Q xPQ
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Rational Functions
The domain of a rational function isthe set of all real numbers except thosethat will make the denominator zero.
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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
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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
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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
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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
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Asymptotes
Consider the graph of
where and are polynomials
with degrees and , respectively.
P xf x
Q x
P x Q x
n m
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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
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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
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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.
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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
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e. 1Rng f
2x
1y
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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
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e. 4Rng g 2, 4
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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
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1e. 0,
2Rng h
1x
0y 1,0.5
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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
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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
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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
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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
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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
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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
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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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Conditional Functions
1
2
Form
condition 1condition 2
condition n
f xf x
f x
f x n
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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
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Example 2.2.19
For the following items,a. find the domainb. find the rangec. sketch the graph
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3 2 if 11.
2 if 1x x
f xx
Dom f
x 0 -2/3y 2 0
1,5
5Rng f
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2
2
1 if 02.
3 1 if 0
1 if 0
x xg x
x x
Dom g
y x x
Rng g
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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
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Absolute Value Functions
Consider
if 0if 0
0,
y f x x
x xy f x x
x x
Dom f
Rng f
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if 0if 0
x xy f x x
x x
0,
Dom f
Rng f
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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
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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
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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