Quiz, week 6
Let
A
=
(
0
,
1
)
A = (0, 1)
. What is
f
(
A
)
f(A)
when
f
(
x
)
=
x
f(x) = x
f
(
x
)
=
(
x
,
y
)
f(x) = (x, y)
f
(
x
)
=
(
x
²
,
y
²
)
f(x) = (x², y²)
f
(
x
)
=
(
cos
π
x
,
sin
π
x
)
f(x) = (\cos πx, \sin πx)
f
:
ℝ
+
+
→
ℝ
f : \mathbb{R}_{++} → \mathbb{R}
is the Bertrand competition example, namely
f
(
x
)
=
{
1
x
if
x
<
1
,
1
2
if
x
=
1
,
0
if
x
>
1
.
f(x) = \begin{cases} \tfrac{1}{x} & \text{if } x < 1, \\ \tfrac{1}{2} & \text{if } x = 1, \\ 0 & \text{if } x > 1. \end{cases}
Let
A
=
(
0
,
1
)
A = (0, 1)
. What is
f
−
1
(
A
)
f^{-1}(A)
when
f
(
x
)
=
x
f(x) = x
f
(
x
,
y
)
=
x
²
+
y
²
f(x, y) = x² + y²
f
(
x
,
y
)
=
x
f(x, y) = x
f
f
is the Bertrand competition example (see above)
Find a disconnected metric space
(
X
,
d
₂
)
(X, d₂)
such that
A
A
is both open and closed, when
A
=
[
0
,
1
]
A = [0, 1]
A
=
(
0
,
1
)
A = (0, 1)
A
=
[
0
,
1
]
×
(
0
,
1
)
A = [0, 1] × (0, 1)