Power point slides

1.   
   
  
  
   
2
5. , 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
 
  
  
 


2.   
   
  
   
  
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
 




3.   
 
2
2 2
11. , 4 2
12. , 1
4 9
function
not a function
x y y x
y x
x y
  
  
  
  

4. Notations
 
 
    

If is in a function then
we say that .
can be replaced ., ,
,
by
fx y
y f x
x y x f x

5. 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
  
 
 
  
 

6. 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.

7. -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 represents
a function.
1.
function

8. -4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y2.
function

9. -4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y3.
not a
function

10. Algebraic Functions
can be obtained by a finite combination
of constants and variables together with
the four basic operations, exponentiation,
or root extractions.

11. Transcendental Functions
those that are not algebraic

12. Polynomial Functions
  1
1 1 0
General Form:
...
Domain:
If 0, the polynomial function is
said to be of degree .
n n
n n
n
y f x a x a x a x a
a f
n

     

¡

13. Constant Functions
 
 
   
Form:
, where is a real number.
Graph: Horizontal Line
y f x C C
Dom f
Rng f C
 


¡

14. -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



¡

15. Linear Functions
 Form:
where and are real numbers, 0
Domain:
Range:
Graph: Line
y f x mx b
m b m
  

¡
¡

16. -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

17. Quadratic Functions
  2
2
Form 1:
Graph is a parabola.
0: opening upward
0: opening downward
4
Vertex: , or ,
2 4 2 2
y f x ax bx c
a
a
b ac b b b
f
a a a a
   


       
    
   

18. 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
b
x
a
Dom f
ac b
Rng f y y a
a
ac b
y y a
a

19. 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 44
vertex: , 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
  
       
  
 
   

 

¡

20. -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

 
¡

21. Quadratic Functions
   
 
2
Form 2:
vertex: ,
y f x a x h k
h k
   

22. 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
  
   
 

  
 
¡

23. -4 -3 -2 -1 1 2 3 4
-3
-2
-1
1
2
3
4
5
6
7
x
y
   
 
2
2 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

  
¡

24. Maximum/Minimum Value
  2
2
2
If ,
4
vertex: ,
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 b
f
a
  
  
 
 



25. Maximum/Minimum Value
  2
2
2
If ,
4
vertex: ,
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 b
f
a
  
  
 
 



26. Example 2.2.7
   
 
   


2
If 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

27. Cubic Functions
   
 
 
3
Form: y f x a x h k
Dom f R
Rng f R
   



28. 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
 
 
 
   
3
Consider
, 0,0
f x x
Dom f R
Rng f R
h k





29. x 1 2 3
y 4 3 2
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
6
7
x
y
Example 2.2.9
   
 
 
   
3
Consider 3 2
, 2,3
f x x
Dom f R
Rng f R
h k
  




30. Rational Functions
 
 
 
Form:
, are polynomials in
degree of 0
degree of 1
P x
y f x
Q x
P Q x
P
Q
 



31. 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

32. 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 

33. 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
  
      
  
   
 

34. 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

35. 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
  
  
     
  
  
 

36. Example 2.2.17
 
   
   
2
g 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

37. 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
1
5.
3
ne
f x x
g x x
h x
j x x
x
k x
 
  

  



38. Conditional Functions
 
 
 
 
1
2
Form
condition 1
condition 2
conditionn
f x
f x
f x
f x n



 



M M

39. 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
  

    
  
   
  
    

40. Example 2.2.19
For the following items,
a. find the domain
b. find the range
c. sketch the graph

41. -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 1
1.
2 if 1
x x
f x
x
Dom f
 
 

 ¡
x 0 -2/3
y 2 0
 1,5
   5Rng f  ¡

42. -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 0
2.
3 1 if 0
1 if 0
x x
g x
x x
Dom g
y x x
  
 
 

   
¡
 Rng g  ¡

43. Absolute Value Functions
 
 
 
   
Consider
if 0
if 0
0,
y f x x
x x
y f x x
x x
Dom f
Rng f
 

   
 

 
¡

44.  
if 0
if 0
x x
y 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

 
¡

45. 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
   

  
  
¡

46. -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

47. -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
7
3 2
3
7
vertex: ,2
3
2
g x x
x
x
Dom g
Rng g y y
  
   
   
 
 
 

 
¡
x 0 3
y -5 0