white (taw933) – HW01 – benzvi – (55600)
2
003
10.0 points
Which, if any, of the following statements are
true when
a, b
are real numbers?
A.
For all positive
a
and
b
,
√
a
+
√
b
=
±
a
+2
√
ab
+
b.
B.
For all positive
a
and
b
,
a

b
√
a
+
√
b
=
√
a

√
C.
For all
a
and
b
,
±
(
a
+
b
)
2
=
a
+
1.
BandCon
ly
2.
Bon
3.
Con
4.
AandBonly
correct
5.
AandConly
6.
none of them
7.
all of them
8.
Aonly
Explanation:
A. TRUE: by the known product,
(
x
+
y
)
2
=
x
2
xy
+
y
2
.
On the other hand,
±
(
x
+
y
)
2
=

x
+
y

,
so if
x
+
y>
0,
x
+
y
=
²
x
2
xy
+
y
2
.
But if
a, b
are positive we can set
x
=
√
a
and
y
=
√
b
.Th
er
e
su
l
tf
o
l
l
ow
ss
in
c
e
x
and
y
are
then positive.
B.
TRUE: by the known diFerence of
squares factorization,
x
2

y
2
=(
x

y
)(
x
+
y
)
.
But if
a, b
are positive we can set
x
=
√
a
and
y
=
√
b
.Theresu
ltfo
l
lowsa
f
terd
iv
is
ion
.
C. ±ALSE:
±
(
x
+
y
)
2
=

x
+
y

,
and since
²
(
·
)i
sa
lw
ay
snon
n
ega
t
e
,th
e
right hand side has to be nonnegative. But
if
a, b
can be positive or negative, an absolute
value sign is then needed on the right.
keywords: square root, properties of square
root, PlaceUT, True±alse, T/±,
004
10.0 points
By removing absolute values, express
f
(
x
)=
x
2

4

x

2

as a piecewisede²ned function.
1.
f
(
x
³
x

2
,x
>

2
,
2

x,
x <

2
.
2.
f
(
x
³
x

2
>
2
,
2

2
.
3.
f
(
x
³
x
>

2
,

(
x
+2)
<

2
.
4.
f
(
x
³
2

x >

2
,
x

2
<

2
.
5.
f
(
x
³

(
x
>
2
,
x
<
2
.
6.
f
(
x
³
x
>
2
,

(
x
<
2
.
cor
rect