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2
Functional Equation
Functional Equation
Source: Israel Autumn 2016 TST2/2
September 29, 2016
algebra
functional equation
Problem Statement
Find all
f
:
R
→
R
f:\mathbb{R}\rightarrow \mathbb{R}
f
:
R
→
R
satisfying (for all
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
):
f
(
x
+
y
)
2
−
f
(
2
x
2
)
=
f
(
y
−
x
)
f
(
y
+
x
)
+
2
x
⋅
f
(
y
)
f(x+y)^2 - f(2x^2) = f(y-x)f(y+x) + 2x\cdot f(y)
f
(
x
+
y
)
2
−
f
(
2
x
2
)
=
f
(
y
−
x
)
f
(
y
+
x
)
+
2
x
⋅
f
(
y
)
.
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