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12
Nice FE over R+
Nice FE over R+
Source: collect
July 19, 2022
functional equation
algebra
Problem Statement
Let
R
+
\mathbb{R}^+
R
+
denote the set of positive real numbers. Find all functions
f
:
R
+
ā
R
+
f:\mathbb{R}^+ \to \mathbb{R}^+
f
:
R
+
ā
R
+
such that
x
+
f
(
y
f
(
x
)
+
1
)
=
x
f
(
x
+
y
)
+
y
f
(
y
f
(
x
)
)
x+f(yf(x)+1)=xf(x+y)+yf(yf(x))
x
+
f
(
y
f
(
x
)
+
1
)
=
x
f
(
x
+
y
)
+
y
f
(
y
f
(
x
))
for all
x
,
y
>
0.
x,y>0.
x
,
y
>
0.
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