For each of the mappings
f
given in Exercise 1, determine whether
f
has a left inverse.
Exhibit a left inverse whenever one exists.
For each of the following mappings
f
:
Z
→
Z
,
exhibit a right inverse of
f
with respect to mapping composition whenever one exists.
a.
f
(
x
)
=
2
x
b.
f
(
x
)
=
3
x
c.
f
(
x
)
=
x
+
2
d.
f
(
x
)
=
1
−
x
e.
f
(
x
)
=
x
3
f.
f
(
x
)
=
x
2
g.
f
(
x
)
=
{
x
if
x
is even
2
x
−
1
if
x
is odd
h.
f
(
x
)
=
{
x
if
x
is even
x
−
1
if
x
is odd
i.
f
(
x
)
=
|
x
|
j.
f
(
x
)
=
x
−
|
x
|
k.
f
(
x
)
=
{
x
if
x
is even
x
−
1
2
if
x
is odd
l.
f
(
x
)
=
{
x
−
1
if
x
is even
2
x
if
x
is odd
m.
f
(
x
)
=
{
x
2
if
x
is even
x
+
2
if
x
is odd
n.
f
(
x
)
=
{
x
+
1
if
x
is even
x
+
1
2
if
x
is odd