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`x^(2)` + bx +
c

We'll replace x by the sum x +
1/x

f(x + 1/x) = a`(x + 1/x)^(2)` + b(x + 1/x) +
c

We'll expand the square:

`(x
+ 1/x)^(2)` = `x^(2)` + 1/`x^(2)` + 2

The function will
become:

f(x + 1/x) = a(`x^(2)` + 1/`x^(2)` + 2) + b(x +
1/x) + c

Since, from enunciation, we'll have f(x + 1/x) =
`x^(2)` + 1/`x^(2)` :

a(`x^(2)` + 1/`x^(2)` + 2) + b(x +
1/x) + c = `x^(2)` + 1/`x^(2)`

Comparing both sides, we'll
get:

a= 1

b =
0

c = -2

1*(`x^(2)` +
1/`x^(2)` + 2) + 0*(x + 1/x) - 2 = `x^(2)` +
1/`x^(2)`

`x^(2)` + 1/`x^(2)` + 2 - 2 = `x^(2)` +
1/`x^(2)`

Therefore, the requested function
f(x) is: f(x) = `x^(2)` - 2.