Lecture Slides on functions

1. FUNCTIONS

2. OBJECTIVES:
• define functions;
• distinguish between dependent and
independent variables;
• represent functions in different ways;
and
• evaluate functions
• define domain and range of a function;
and
• determine the domain and range of a
function

3. 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.
• A function is a correspondence from a set X of
real numbers x to a set Y of real numbers y,
where the number y is unique for a specific
value of x.

4. 
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

5. 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 not all
relation is a function.

6. In order to have a function, there must be
one value of the dependent variable (y) for
each value of the independent variable (x).
Or, there could also be two or more
independent variables (x) for every dependent
variable (y). These correspondences are
called one-to-one correspondence and many-
to-one correspondence, respectively.
Therefore, a function is a set of ordered pairs
of numbers (x, y) in which no two distinct
ordered pairs have the same first number.

7. Ways of Expressing a function
5. Mapping
2. Tabular form
3. Equation
4. Graph
1. Set notation

8. .
Example: Express the function 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

9. 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

10. EXAMPLE:
Determine whether or not each of the following
sets represents a function:
1. A = {(-1, -1), (10, 0), (2, -3), (-4, -1)}
2. B = {(2, a), (2, -a), (2, 2a), (3, a2)}
3. C = {(a, b)| a and b are integers and a = b2}
4. D = {(a, b)| a and b are positive integers and a = b2}
5.  
 
4
x
y
|
y
,
x
E 2




11. There are more than one element as the first
component of the ordered pair with the same
second component namely (-1, -1) and (-4, -1),
called a many-to-one correspondence. One-to-
many correspondence is a not function but
many-to-one correspondence is a function.
There exists one-to-many correspondence
namely, (2, a), (2, -a) and (2, 2a).
SOLUTIONS:
1. A is a function.
2. B is a not a function.

12. 3. C is not a function.
There exists a one-to-many correspondence in
C such as (1, 1) and (1, -1), (4, 2) and (4, -2),
(9, 3) and (9, -3), etc.
4. D is a function.
The ordered pairs with negative values in solution
c above are no longer elements of C since a and b
are given as positive integers. Therefore, one-to-
many correspondence does not exist anymore in
set D.
5. E is not a function
Because for every value of x, y will have two
values.

13. OTHER EXAMPLES:
Determine whether or not each of the following sets
represents a function:
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)

14. DEFINITION: FUNCTION NOTATION
• Letters like f , g , h, 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 ”.

15. 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))
   2
2
2
)
(
,
)
(
),
1
(
),
(
),
0
(
),
3
( h
a
f
a
f
a
f
a
f
f
f 

5. If find each of the following
1
)
( 2

 x
x
f

16. h
)
x
(
g
)
h
x
(
g 

6. Find (a) g(2 + h), (b) g(x + h), (c)
where h 0 if
.
1
x
x
)
x
(
g


7. Given that show that
.
,
1
)
(
x
x
F 
kx
x
k
x
F
k
x
F




 2
)
(
)
(
3
2
3
4
)
,
( y
xy
x
y
x
f 

 )
,
(
)
,
( 3
y
x
f
a
ay
ax
f 
8. If , show that
,
)
,
(
v
u
v
u
v
u
f


  
v
u
f
v
u
f ,
1
,
1







9. If find

17. 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 functions.
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}

18. 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
)
,
( 
 )
,
0
[ 
)
,
( 
 )
,
5
[ 


19. 2
x
x
2
y


4.
Answer:
D: all real nos. except -2
R: all real nos. except 2
1
x
y 

5.
Answer :
D: all real nos. > –1
R: all real nos. > 0
3
x
x
3
y



6.
Answer:
D: all real nos. <3
R: all real nos. <0
2
except
)
,
(
:
D 

 2
)
,
(
: 

 except
R
)
,
1
[
:
D 
 )
,
0
[
:
R 
)
3
,
(
:
D   
0
,
:
R 


20. From the above examples, you can draw conclusions and
formulate the following theorems on the domain
determination of functions.
Theorem 1. The domain of a polynomial function is the set
of all real numbers or (-, +).
Theorem 3. A rational function f is a ratio of two polynomials:
The domain of a rational function consists of all values of x
such that the denominator is not equal to zero
where P and Q are polynomials.
)
x
(
Q
)
x
(
P
)
x
(
f 
Theorem 2. The domain of is the set of all real
numbers satisfying the inequality f(x)  0 if n is even integer
and the set of all real numbers if n is odd integer.
n
)
x
(
f

21. 1.An algebraic function is the result when the
constant function, (f(x) = k, k is constant) and the
identity function (g(x) = x) are put together by
using a combination of any four operations, that is,
addition, subtraction, multiplication, division, and
raising to powers and extraction of roots.
KINDS OF FUNCTIONS:
Example: f(x) = 5x – 4,
4
x
7
x
2
x
)
x
(
g 2




22. Generally, functions which are not classified as
algebraic function are considered as transcendental
functions namely the exponential, logarithmic,
trigonometric, inverse trigonometric, hyperbolic and
inverse hyperbolic functions.

23. 7
3
1



y
x
A. Which of the following represents a function?
1.A = {(2, -3), (1, 0), (0, 0), (-1, -1)}
2.B = {(a, b)|b = ea}
3.C = {(x, y)| y = 2x + 1}
4.
5. E = {(x, y)|y = (x -1)2 + 2}
6. F = {(x, y)|x = (y+1)3 – 2}
7. G = {(x, y)|x2 + y2 = }
8. H = {(x, y)|x  y}
9. I = {(x, y)| |x| + |y| = 1}
10. J = {(x, y)|x is positive integer and
EXERCISES:
 
 
2
1
|
, a
b
b
a
D 



24. B. Given the function f defined by f(x) = 2x2 + 3x – 1,
find:
a. f(0) f. f(3 – x2)
b. f(1/2) g. f(2x3)
c. f(-3) h. f(x) + f(h)
d. f(k + 1) i. [f(x)]2 – [f(2)]2
e. f(h – 1) j. 0
h
;
h
)
x
(
f
)
h
x
(
f




25. ,
3
2
)
( 
 x
x
F
C. Given find
0
;
)
(
)
(
.
5
,
2
1
.
4
),
3
2
(
.
3
),
4
(
.
2
),
1
(
.
1











h
h
x
F
h
x
F
F
x
F
F
F

26. EXERCISES:
Find the domain and range of the following functions:
2
2
2
2
4
:
.
6
1
1
2
)
(
.
5
3
)
(
.
4
2
1
)
(
.
3
1
)
(
.
2
3
4
)
(
.
1
x
y
g
x
x
x
x
f
x
x
h
x
x
G
x
x
F
x
x
f














  
  
3
12
9
4
3
:
.
10
4
:
.
9
3
1
3
)
(
.
8
1
2
1
)
(
.
7
2
2
2























x
x
x
x
x
x
y
H
x
y
g
x
f
x
x
x
F
if
if
3
3


x
x
if
if
if
2
2
1
1




x
x
x

27. Exercises: Identify the domain and range of the
following functions.
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.