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If \(\rm f(x-\frac{1}{x})=x^2+\frac{1}{x^2}\), then f(x) = ?
1. x2 - 1
2. x2 + 1
3. x2 - 2
4. x2 + 2 

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Best answer
Correct Answer - Option 4 : x2 + 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. 

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