Since the result of f(x+1/x) is a quadratic function, then
f(x) is a quadratic function, too.
If f(x) is a quadratic
function, we'll have:
f(x) = a + bx +
c
We'll replace x by the sum x +
1/x
f(x + 1/x) = a + b(x + 1/x) +
c
We'll expand the square:
+ 1/x)^(2)x^(2)
x^(2)
The function will
become:
f(x + 1/x) = a( + 1/
+ 2) + b(x +
1/x) + c
Since, from enunciation, we'll have f(x + 1/x) =
+ 1/
:
a( + 1/
+ 2) + b(x +
1/x) + c = + 1/
Comparing both sides, we'll
get:
a= 1
b =
0
c = -2
1*( +
1/ + 2) + 0*(x + 1/x) - 2 =
+
1/
+ 1/
+ 2 - 2 =
+
1/
Therefore, the requested function
f(x) is: f(x) = - 2.
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