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Mathematics
Problem
Let
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
be a function such 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
, and
g
:
R
→
(
0
,
∞
)
g: \mathbb{R} \to (0, \infty)
g
:
R
→
(
0
,
∞
)
be a function such that
g
(
x
+
y
)
=
g
(
x
)
g
(
y
)
g(x+y) = g(x)g(y)
g
(
x
+
y
)
=
g
(
x
)
g
(
y
)
for all
x
,
y
∈
R
x, y \in \mathbb{R}
x
,
y
∈
R
. If
f
(
−
3
5
)
=
12
f\left(\frac{-3}{5}\right) = 12
f
(
5
−
3
)
=
12
and
g
(
−
1
3
)
=
2
g\left(\frac{-1}{3}\right) = 2
g
(
3
−
1
)
=
2
, then the value of
(
f
(
1
4
)
+
g
(
−
2
)
−
8
)
g
(
0
)
\left(f\left(\frac{1}{4}\right) + g(-2) - 8\right)g(0)
(
f
(
4
1
)
+
g
(
−
2
)
−
8
)
g
(
0
)
is _____.
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