Correct Answer - Option 4 : x
2 + 2
Concept:
\(\rm (a-b)^2=a^2-2ab+b^2\)
Calculation:
Given: \(\rm f(x-\frac{1}{x})=x^2+\frac{1}{x^2}\)
Let, \(\rm x-\frac1x\) = y
∴ \(\rm f(x-\frac{1}{x})=f(y) =x^2+\frac{1}{x^2}\)
\(\rm ⇒ f(y) =x^2+\frac{1}{x^2}-2+2\)
\(\rm =(x-\frac{1}{x})^2+2\) (∵ \(\rm x^2+\frac{1}{x^2}-2 = (x-\frac1x)^2\) )
= y2 + 2 (∵ \(\rm x-\frac1x= y\))
⇒ f(y) = y2 + 2
⇒ f(x) = x2 + 2 (replace y by x)
Hence, option (4) is correct.