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3
Prove f(x+y+xy)=f(x)+f(y)+f(xy) implies f(x+y)=f(x)+f(y)
Prove f(x+y+xy)=f(x)+f(y)+f(xy) implies f(x+y)=f(x)+f(y)
Source:
July 17, 2012
function
algebra proposed
algebra
Problem Statement
Let
f
:
R
⟶
R
f:\mathbb{R}\longrightarrow \mathbb{R}
f
:
R
⟶
R
be a function such that
f
(
x
+
y
+
x
y
)
=
f
(
x
)
+
f
(
y
)
+
f
(
x
y
)
f(x+y+xy)=f(x)+f(y)+f(xy)
f
(
x
+
y
+
x
y
)
=
f
(
x
)
+
f
(
y
)
+
f
(
x
y
)
for all
x
,
y
∈
R
x, y\in\mathbb{R}
x
,
y
∈
R
. Prove that
f
f
f
satisfies
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
.
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