Monday, February 20, 2012

function satisfies condition f(x+1/x)=x^2+1/x^2 . find function

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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