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ISI B.Stat Entrance Exam
2010 ISI B.Stat Entrance Exam
5
(g(h(x))=h(g(x))
(g(h(x))=h(g(x))
Source: ISI(BS) 2010 #5
May 17, 2012
function
algebra proposed
algebra
Problem Statement
Let
A
A
A
be the set of all functions
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
f
(
x
y
)
=
x
f
(
y
)
f(xy)=xf(y)
f
(
x
y
)
=
x
f
(
y
)
for all
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
.(a) If
f
∈
A
f \in A
f
∈
A
then show that
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
f(x+y)=f(x)+f(y)
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
for all
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
(b) For
g
,
h
∈
A
g,h \in A
g
,
h
∈
A
, define a function
g
∘
h
g\circ h
g
∘
h
by
(
g
∘
h
)
(
x
)
=
g
(
h
(
x
)
)
(g \circ h)(x)=g(h(x))
(
g
∘
h
)
(
x
)
=
g
(
h
(
x
))
for
x
∈
R
x \in \mathbb{R}
x
∈
R
. Prove that
g
∘
h
g \circ h
g
∘
h
is in
A
A
A
and is equal to
h
∘
g
h \circ g
h
∘
g
.
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