The document discusses functions and relations. It defines functions, relations, and domain and range. It provides examples of expressing relations in set notation, tabular form, equations, graphs, and mappings. It also discusses evaluating, adding, multiplying, dividing, and composing functions. Graphs of various functions like absolute value, piecewise, greatest integer, and least integer functions are also explained.

1. Functions
Prepared by:
Malik Sabah-ud-din
1
Basic2advanced.blogspot.com

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
2
Basic2advanced.blogspot.com

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
3
Basic2advanced.blogspot.com

4. Ways of Expressing a Relation
5. Mapping
2. Tabular form
3. Equation
4. Graph
1. Set notation
4
Basic2advanced.blogspot.com

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
5
Basic2advanced.blogspot.com

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
6
Basic2advanced.blogspot.com

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}
7
Basic2advanced.blogspot.com

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
[  )
,
( 

8
Basic2advanced.blogspot.com

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 

9
Basic2advanced.blogspot.com

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