Question

If f is a real function satisfying $f\left(x+\frac{1}{x}\right)=x^2+\frac{1}{x^2}$for all x ∈ R − {0}, then write the expression for f(x).

Answer 1

Given:
$f\left(x+\frac{1}{x}\right)=x^2+\frac{1}{x^2}$
$=x^2+\frac{1}{x^2}+2-2$
$={\left(x+\frac{1}{x}\right)}^2-2$
Thus,
$f\left(x+\frac{1}{x}\right)={\left(x+\frac{1}{x}\right)}^2-2$
Hence,
f (x) = x² $-$ 2 , where | x | ≥ 2.