MATH 106 CALCULUS I FOR BIO. & SOC. SCI. FALL 2011
INSTRUCTOR: JOS
´
E MANUEL G
´
OMEZ
Solutions Homework 1.
Please show all your work.
(1) (5 Points) The functions
f
and
g
are defined below
f
(
x
) =
√
x
2
−
1
,
g
(
x
) =
1
x
.
Find explicit descriptions and the domains of the following functions.
(a)
f
◦
g
,
(b)
g
◦
f
,
(c)
f
◦
f
,
(d)
g
◦
g
.
Solution:
Let’s start by finding the domains of
f
and
g
. Clearly the domain of
g
are all numbers
x
negationslash
= 0. On the other hand, the domain of
f
is the set of numbers
x
such that
x
2
−
1
≥
0; that is
x
2
≥
1. This shows that

x

=
√
x
2
≥
1. The solution
of this inequality is the set of elements
x
≥
1 and
x
≤ −
1.
(a)
f
◦
g
(
x
) =
f
(
g
(
x
)) =
f
parenleftbigg
1
x
parenrightbigg
=
radicalbigg
1
x
2
−
1
.
The domain of
f
◦
g
is the set of elements
x
for which
g
(
x
) is defined and
g
(
x
) is in
the domain of
f
. This shows that
x
negationslash
= 0 and that

g
(
x
)

=
1

x

≥
1. This shows that

x
 ≤
1. It follows that the domain of
f
◦
g
is [
−
1
,
0)
∪
(0
,
1].
(b)
g
◦
f
(
x
) =
g
(
f
(
x
)) =
g
parenleftBig
√
x
2
−
1
parenrightBig
=
1
√
x
2
−
1
.
The domain of
g
◦
f
is the set of elements for which
f
(
x
) =
√
x
2
−
1 makes sense and
f
(
x
) is in the domain of
g
; that is,
f
(
x
)
negationslash
= 0. So we need that
x
≥
1 or
x
≤ −
1 and
we also need
√
x
2
−
1
negationslash
= 0. Note that
√
x
2
−
1 = 0 precisely when
x
2
= 1; that is,
x
=
±
1. We conclude that the domain of
g
◦
f
is (
−∞
,
−
1)
∪
(1
,
∞
).
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 Fall '11
 JoseGomez
 Calculus, Radioactive Decay, Radionuclide, Isotope, JOSE MANUEL GOMEZ

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