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Section 3.1
55
17. (a) The function is not onto since, for example,
g(x)
=
0 has no integer solutions (by the quadratic
formula), but it is onetoone. To see why, suppose
g(X1)
=
g(X2).
Then
3x~
+
14x1

51
=
3x~
+
14x2
 51, so
3(X1

X2)(X1
+
X2)
=
14(x2

Xl).
Thus
Xl
=
X2
or
Xl
+
X2
=
li.
Since
Xl
and
X2
are integers,
Xl
+
X2
= 
13
4
is impossible. Thus,
Xl
=
X2,
so
9
is onetoone.
(b) Since the graph of
9
is a parabola,
9
is neither onetoone nor onto.
18. (a) [BB] Note that
f(x)
=
(x
+
7)2 
100.
If
f(X1)
=
f(X2)
it follows that
(Xl
+
7)2 
100
=
(X2
+
7)2 
100, so
(Xl
+
7)2
=
(X2
+
7)2
and, taking square roots,
IX1
+
71
=
IX2
+
71.
Since
Xl, X2
E N,
we know
Xl
+
7
>
0
and
X2
+
7
>
O.
Thus,
Xl
+
7 =
X2
+
7
and
Xl
=
X2.
Thus,
f
is onetoone, but it is not onto: For example,
1 E
B,
but there is no
X
E N
with
1
(x)
= 1
since
(x
+
7)2 
100
=
1 implies
(x
+
7)2 =
101 and this equation has no solution in the natural
numbers.
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Quadratic Formula, Graph Theory

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