Section
3.3
69
x
=
!.
If
r
=1=
0,
the quadratic formula tells us that this equation has solutions
r
+
1
±
J(r
+
1)2

2r
r
+
1
±
v'r2
+
1
x
=
2r
=
2r':""'
.
We claim that the solution
r+
1
Jr2
+
1
x
=
:
2r
is always in
(0,1).
First note that
if
r
>
0,
then
r
+
1
>
v'r2
+
1
since
(r
+
1)2
>
r2
+
1.
On the
other hand,
if
r
<
0,
then
r
+
1
<
1
<
v'r2
+
1.
In
either case, we have shown that
r
+
1
v'r2
+
1
2r
>
O.
To show that this expression is also less than 1, we will prove the equivalent inequality
lr
Jr2
+1
2r
<
O.
If
r
>
0,
then
1 
r
<
1
<
v'r2
+
1
giving the result.
If
r
<
0,
then
1 
r
>
v'r2
+
1
because
(1

r)2
>
r2+
1, again giving the result. Since
f
is onetoone and onto,
'tis
a onetoone correspondence.
18. Start at
(0,0)
and move as illustrated.
19.
(a)
2,
2,4,
4,
8,
8,
16,
16,
...
(b)
1=2°,2,!,4,i,8,~,
...
(c)
1,4,7,10,13,16,19,
...
·
ree
·
r
re
• . r r r
•
r r

•
r

•
•
• •
eee
ee
1
e
1 1
•
•
• •
•
(d)
(1,1),
(1,
2), (1,3),
(2,
1),
(2,
2),
(3,
3),
(3,
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 Summer '10
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 Quadratic Formula, Graph Theory, Binary relation, onetoone correspondence, r2 other hand, equivalent inequality

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