Solu tion
As all three elements of the codomain are images of elements in the domain,
we have f as an “onto function”.
Ì Exam ple 0.1.17: Given f x
x
( ) = +
2 3 and
g x
x
( ) = +
3 2
Check if commutative law holds good for composition of functions.
Solu tion
(
)
(
)( )
( )
(
)
(
)
(
)( )
(
f g x
f g x
f x
x
x
g f x
g f x
⋅
=
=
+
=
+ +
=
+
⋅
=
3 2
2 3 2 3
6 7
(
)
)
(
)
(
)
=
+
=
+ +
=
+
g x
x
x
2 3
3 2 3 2
6 11
Since (
)( ) (
)( )
f g x
g f x
⋅
≠ ⋅
, commutative law does not hold for composition
of functions.
Ì Exam ple 0.1.18: Check whether the mapping f X
X
: →
where
X
x R x
= ∈
≠
{
,
}
0 defined by f x
x
( ) =
1 is one to one and onto.
Solu tion
x = set of all non-zero real numbers. Let x x
X
1
2
, ∈ .
Then
f x
f x
x
x
x
x
( )
( )
1
2
1
2
1
2
1
1
=
⇒
=
⇒ =
Hence f is one-to-one.
14
Theory of Automata, Formal Languages and Computation
a
b
c
d
1
2
3
A
B