if f(x + \frac{1}{x}) = x^2 + \frac{1}{x^2} then f(x) = ? | Mathematics - Functional Equations | SolveeIt

if $f(x + \frac{1}{x}) = x^2 + \frac{1}{x^2}$ then f(x) = ?

$(x^2 - 2)$

$(x^2 + 1)$

$(x^2 - 1)$

$x^2$

Answer

$(x^2 - 2)$

Explanation

Let

$u = x + \frac{1}{x}$ .

Then

$u^2 = \left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} \quad \Rightarrow \quad x^2 + \frac{1}{x^2} = u^2 - 2$ .

Since the given equation is

$f\left(x+\frac{1}{x}\right)=x^2+\frac{1}{x^2}$ ,

we substitute $u$ to obtain:

$f(u)= u^2 - 2$ .

Thus, replacing $u$ by $x$ (as the function variable), we get:

$f(x) = x^2 - 2$ .

Question: if $f(x + \frac{1}{x}) = x^2 + \frac{1}{x^2}$ then f(x) = ?...