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68
Solutions to Exercises
(b) [BB] The function
f:
(2,5) 
(0,1) defined by
f(x)
=
ix

~
is a onetoone correspondence,
as is the function
g:
(0,1) 
(10,00) defined by
g(x)
= .!.
+
9. Thus, the composition
go
x
f:
(2,5) 
(10,00) is also a onetoone correspondence. Note that
(g
0
f)(x)
=
~2
+
9.
x
(c) The function
f: (a, b)

(0,1) defined by
f(x)
=
xb

a
is a onetoone correspondence, as is
a
the function
g:
(0, 1) 
(c,
00) defined by
g(x)
= .!.

1
+
c. Thus a onetoone correspondence
x
(a, b)
_
(c,
00) is the composition
9
0
f.
Note that
(g
0
f)(x)
=
b

a
 1
+
c.
xa
14. In Problem 27, p. 129, we saw that the function
g:
(0,1) 
(0,00) defined by
g(x)
=
.!.

1 is a
x
onetoone correspondence. Furthermore, the function
f: (a, b)
_ (0,1) defined by
f(x)
=
xb

a
is
a
a onetoone correspondence. (See the remarks preceding Problem 28.) Thus
9
0
f:
(a,
b)

(0,00)
ba
is a onetoone correspondence. Note that
(g
0
f)(x)
=

l.
xa
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 Summer '10
 any
 Graph Theory

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