chapter 2.2 algebraic functions. definition of functions
TRANSCRIPT
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