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.