Find the Expression for \( f(x) \)
Question:
If \(f\) is a real function satisfying
\[ f\left(x+\frac1x\right) = x^2+\frac1{x^2} \]
for all
\[ x\in R-\{0\}, \]
then write the expression for \(f(x)\).
Solution:
Let
\[ t=x+\frac1x \]
Squaring,
\[ t^2=x^2+\frac1{x^2}+2 \]
\[ x^2+\frac1{x^2}=t^2-2 \]
Therefore,
\[ f(t)=t^2-2 \]
Replacing \(t\) by \(x\),
\[ \boxed{f(x)=x^2-2} \]