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56
Solutions to Exercises
20.
(a) [BB] The graph of
1
shown at the right makes it clear that
1
is
onetoone and onto.
(b) [BB] Solution 1. Since
I:
R + R is onetoone by part (a), it is
also onetoone as a function with domain Z. Here, however,
1
is
not onto for we note that
1(0)
=
0,
1(1)
=
4 and
1
is increasing,
so 1 is not in the range of
I.
X
Solution 2. (This solution mimics that given in Problem 8 in our discussion of discrete functions in
this section.)
If
I(xt}
=
I(X2),
then
3xf+xI
=
3X~+X2'
so
3(xfx~)
=
X2XI
and
3(XIX2)(X~+XIX2+X~)
=
X2

Xl.
If
Xl
f:.
X2,
we must have
x~
+
XIX2
+
x~
=
i,
which is impossible for integers Xl,
X2.
Thus, Xl
=
X2
and
1
is onetoone.
On the other hand,
1
is not onto. In particular, 1 is not in the range of
1
since 1
=
1
(
a)
for some
a
implies
3a
3
+
a
=
1; that is,
a(3a
2
+
1)
=
1. But the only pairs of integers whose product is 1 are the
pairs 1,1 and
1,
1. So here, we'd have to have
a
=
3a
2
+
1
=
1 or
a
=
3a
2
+
1
=
1, neither of
which is possible.
21. (a) The graph of
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 Summer '10
 any
 Graph Theory

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