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2011 Greece Team Selection Test
3
Find two functions from two conditions
Find two functions from two conditions
Source: Greek TST 2011-pr3
May 23, 2016
function
functional equation
algebra
Problem Statement
Find all functions
f
,
g
:
Q
→
Q
f,g: \mathbb{Q}\to \mathbb{Q}
f
,
g
:
Q
→
Q
such that the following two conditions hold:
f
(
g
(
x
)
−
g
(
y
)
)
=
f
(
g
(
x
)
)
−
y
(
1
)
f(g(x)-g(y))=f(g(x))-y \ \ (1)
f
(
g
(
x
)
−
g
(
y
))
=
f
(
g
(
x
))
−
y
(
1
)
g
(
f
(
x
)
−
f
(
y
)
)
=
g
(
f
(
x
)
)
−
y
(
2
)
g(f(x)-f(y))=g(f(x))-y\ \ (2)
g
(
f
(
x
)
−
f
(
y
))
=
g
(
f
(
x
))
−
y
(
2
)
for all
x
,
y
∈
Q
x,y \in \mathbb{Q}
x
,
y
∈
Q
.
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