1. Functions
Prepared by:
Malik Sabah-ud-din
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2. OBJECTIVES
•distinguish functions and relations
•identify domain and range of a function/relation evaluate
functions/relations.
•perform operation on functions/relations
•graph functions/relations
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3. Relation is referred to as any set of ordered pair.
Conventionally, It is represented by the ordered pair
( x , y ). x is called the first element or x-coordinate
while y is the second element or y-coordinate of the
ordered pair.
DEFINITION
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4. Ways of Expressing a Relation
5. Mapping
2. Tabular form
3. Equation
4. Graph
1. Set notation
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5. .
Example: Express the relation y = 2x;x= 0,1,2,3
in 5 ways.
1. Set notation
(a) S = { ( 0, 0) , ( 1, 2 ) , ( 2, 4 ), ( 3, 6) } or
(b) S = { (x , y) such that y = 2x, x = 0, 1, 2, 3 }
2. Tabular form
x 0 1 2 3
y 0 2 4 6
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6. 3. Equation: y = 2x
4. Graph
y
x
5
-4 -2 1 3 5
5
-4
-2
1
3
5
-5 -1 4
-5
-1
4
-3
-5
2
2
-5
-3
●
●
●
1 2
2 4
6
3
0 0
x y
5. Mapping
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7. DEFINITION: Domain and Range
All the possible values of x is called the domain and all the
possible values of y is called the range. In a set of ordered
pairs, the set of first elements and second elements of
ordered pairs is the domain and range, respectively.
Example: Identify the domain and range of the following
relations.
1.) S = { ( 4, 7 ),( 5, 8 ),( 6, 9 ),( 7, 10 ),( 8, 11 ) }
Answer : D: { 4,5,6,7,8} R:{7,8,9,10,11}
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8. 2.) S = { ( x , y ) s. t. y = | x | ; x  R }
Answer: D: all real nos. R: all real nos. > 0
3) y = x 2 – 5
Answer. D: all real nos. R: all real nos. > -5
4) | y | = x
Answer: D: all real nos. > 0 R: all real nos.
)
,
( 
 )
,
0
[ 
)
,
( 
 )
,
5
[ 

)
,
0
[  )
,
( 

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9. 2
x
x
2
y


5.
Answer:
D: all real nos. except -2
R: all real nos. except 2
1
x
y 

6. Answer :
D: all real nos. > –1
R: all real nos. > 0
g)
3
3



x
x
y
7. Answer:
D: all real nos. < 3
R: all real nos. except 0
2
except
)
,
(
:
D 

 2
except
)
,
(
:
D 


)
,
1
[
:
D 

)
,
0
[
:
R 
)
3
,
(
: 
D
0
except
)
,
(
:
D 

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10. Exercises: Identify the domain and range of the
following relations.
1. {(x,y) | y = x 2 – 4 }
8. y = (x 2 – 3) 2








x
2
x
3
y
)
y
,
x
(
4.
 
3
)
,
( x
y
y
x 
2.
 
9
)
,
( 
 x
y
y
x
3.
 
4
x
3
x
y
)
y
,
x
( 2



5.
y = | x – 7 |
6.
7. y = 25 – x 2
x
5
x
3
y


9.
5
x
25
x
y
2



10.
10

11. PROBLEM SET #5-1
FUNCTIONS
Identify the domain and range of the following relations.
11

12. 12
Definition: Function
•A function is a special relation such that every first
element is paired to a unique second element.
•It is a set of ordered pairs with no two pairs having
the same first element.

13. 
x
y sin

1
3

 x
y
One-to-one and many-to-one functions
Each value of x maps to only one
value of y . . .
Consider the following graphs
Each value of x maps to only one
value of y . . .
BUT many other x values map to
that y.
and each y is mapped from only
one x.
and
Functions
13

14. One-to-one and many-to-one functions
is an example of a
one-to-one function
1
3

 x
y
is an example of a
many-to-one function

x
y sin


x
y sin

1
3

 x
y
Consider the following graphs
and
Functions
One-to-many is NOT a function. It is just a
relation. Thus a function is a relation but a relation
could never be a function.
14

15. 15
Example: Identify which of the following
relations are functions.
a) S = { ( 4, 7 ), ( 5, 8 ), ( 6, 9 ), ( 7, 10 ), ( 8, 11 ) }
b) S = { ( x , y ) s. t. y = | x | ; x  R }
c) y = x 2 – 5
d) | y | = x
2
x
x
2
y


e)
1
x
y 

f)

16. 16
DEFINITION: Function Notation
•Letters like f , g , h and the likes are used to designate
functions.
•When we use f as a function, then for each x in the
domain of f , f ( x ) denotes the image of x under f .
•The notation f ( x ) is read as “ f of x ”.

17. 17
EXAMPLE: Evaluate each function
value
1. If f ( x ) = x + 9 , what is the value of f ( x 2 ) ?
2. If g ( x ) = 2x – 12 , what is the value of g (– 2 )?
3. If h ( x ) = x 2 + 5 , find h ( x + 1 ).
4. If f(x) = x – 2 and g(x) = 2x2 – 3 x – 5 ,
Find: a) f(g(x)) b) g(f(x))

18. 18
Piecewise Defined Function
if x<0





1
x
x
)
x
(
f
.
1
2
0
x 
if 







2
)
2
x
(
1
x
x
3
)
x
(
f
.
2
A piecewise defined function is defined
by different formulas on different parts
of its domain.
Example:

19. 19
Piecewise Defined Function
if x<0 f(-2), f(-1), f(0), f(1), f(2)
EXAMPLE: Evaluate the piecewise function at the
indicated values.





1
x
x
)
x
(
f
.
1
2
0
x 
if








2
)
2
x
(
1
x
x
3
)
x
(
f
.
2
f(-5), f(0), f(1), f(5)
0
x 
if
if
if
2
x
0 

2
x 

20. 20
DEFINITION: Operations on
Functions
If f (x) and g (x) are two functions, then
a) Sum and Difference
( f + g ) ( x ) = f(x) + g(x)
b) Product
( f g ) ( x ) = [ f(x) ] [ g(x) ]
c) Quotient
( f / g ) ( x ) = f(x) / g(x)
d) Composite
( f ◦ g ) ( x ) = f (g(x))

21. Example :1. Given f(x) = 11– x and g(x) = x 2 +2x –10
evaluate each of the following functions
a. f(-5)
b. g(2)
c. (f g)(5)
d. (f - g)(4)
e. f(7)+g(x)
f. g(-1) – f(-4)
g. (f ○ g)(x)
h. (g ○ f)(x)
i. (g ○ f)(2)
j. (f○ g)

)
(x 2
21

22. 
22

23. 
23

24. 24
DEFINITION: Graph of a Function
•If f(x) is a function, then its graph is the set of all points
(x,y) in the two-dimensional plane for which (x,y) is an
ordered pair in f(x)
•One way to graph a function is by point plotting.
•We can also find the domain and range from the
graph of a function.

25. 25
Example: Graph each of the following
functions.
5
x
3
y
.
1 

1
x
y
.
2 

2
x
16
y
.
3 

5
x
y
.
4 2


3
x
2
y
.
5 
x
5
x
3
y


6.
4
x
y
.
7 



26. 26
Graph of piecewise defined function
The graph of a piecewise function consists of separate
functions.





1
x
2
x
)
x
(
f
.
1
2
if
if 1
x 
1
x 









3
x
x
9
x
)
x
(
f
.
2 2
0
x 
1
x 
3
x
0 

if
if
if
Example: Graph each piecewise function.

27. x
y
1
-2
Plot the points in the coordinate plane
27

28. x
y
1
-2
Plot the points in the coordinate plane
28

29. 29
Graph of absolute value function.
Recall that





x
x
x
if
if
0
x 
0
x 
Using the same method that we used in graphing
piecewise function, we note that the graph of f
coincides with the line y=x to the right of the y axis
and coincides with the line y= -x the left of the y-axis.

30. 30
Example: Graph each of the follow
functions.
y = | x – 7 |
1.
y = x-| x - 2 |
4.

31. x
y
1
-2
Plot the points in the coordinate plane
31

32. 
32

33. 33
Definition: Greatest integer function.
greatest integer less than or equal to x
The greatest integer function is defined by
  
x
Example:
  
0
  
1
.
0
  
3
.
0
  
9
.
0
  
1
  
1
.
1
  
2
.
1
  
9
.
1
  
2
  
1
.
2
  
4
.
3
  
 4
.
3
  
 9
.
0
0
0
0
0
1
1
1
1
2
2
3
-4
-1

34. 34
Definition: Least integer function.
least integer greater than or equal to x
The least integer function is defined by
  
x
Example:
  
0
  
1
.
0
  
3
.
0
  
9
.
0
  
1
  
1
.
1
  
2
.
1
  
9
.
1
  
2
  
1
.
2
  
4
.
3
  
 4
.
3
  
 9
.
0
0
1
1
1
1
2
2
2
2
3
4
-3
0

35. 35
Graph of greatest integer function.
 
x
y 
Sketch the graph of
x  
x
y 
1
x
2 



0
x
1 


1
x
0 

2
x
1 

3
x
2 

2

1

0
1
2

36. x
y
1
-2
Plot the points in the coordinate plane
36

37. 37
Graph of least integer function.
 
x
y 
Sketch the graph of
x  
x
y 
1
x
2 



0
x
1 


1
x
0 

2
x
1 

3
x
2 

1

0
1
2
3

38. x
y
1
-2
Plot the points in the coordinate plane
38