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Mathematics
Problem
Which of the following is always true for an invertible function
f
:
X
→
Y
f:X\to Y
f
:
X
→
Y
and its inverse
f
−
1
f^{-1}
f
−
1
? a)
f
−
1
(
f
(
x
)
)
=
x
for all
x
∈
Y
f^{-1}(f(x))=x \text{ for all } x\in Y
f
−
1
(
f
(
x
))
=
x
for all
x
∈
Y
b)
f
(
f
−
1
(
y
)
)
=
y
for all
y
∈
Y
and
Dom
(
f
−
1
)
=
Y
f(f^{-1}(y))=y \text{ for all } y\in Y \text{ and } \operatorname{Dom}(f^{-1})=Y
f
(
f
−
1
(
y
))
=
y
for all
y
∈
Y
and
Dom
(
f
−
1
)
=
Y
c)
Dom
(
f
−
1
)
=
X
and
Ran
(
f
−
1
)
=
Y
\operatorname{Dom}(f^{-1})=X \text{ and } \operatorname{Ran}(f^{-1})=Y
Dom
(
f
−
1
)
=
X
and
Ran
(
f
−
1
)
=
Y
d)
f
−
1
(
y
)
=
1
f
(
y
)
f^{-1}(y)=\dfrac{1}{f(y)}
f
−
1
(
y
)
=
f
(
y
)
1
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