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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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.

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