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If F ( X ) = X 3 − 1 X 3 , Show that F ( X ) + F ( 1 X ) = 0 . - Mathematics

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Question

If  \[f\left( x \right) = x^3 - \frac{1}{x^3}\] , show that

\[f\left( x \right) + f\left( \frac{1}{x} \right) = 0 .\]
 

 

Solution

Given:

\[f\left( x \right) = x^3 - \frac{1}{x^3}\]    ...(i)
Thus,
\[f\left( \frac{1}{x} \right) = \left( \frac{1}{x} \right)^3 - \frac{1}{\left( \frac{1}{x} \right)^3}\] \[= \frac{1}{x^3} - \frac{1}{\frac{1}{x^3}}\]
\[\therefore f\left( \frac{1}{x} \right) = \frac{1}{x^3} - x^3\]  ...(ii) 
\[f\left( x \right) + f\left( \frac{1}{x} \right) = \left( x^3 - \frac{1}{x^3} \right) + \left( \frac{1}{x^3} - x^3 \right)\]
\[= x^3 - \frac{1}{x^3} + \frac{1}{x^3} - x^3 = 0\] 
Hence,
\[f\left( x \right) + f\left( \frac{1}{x} \right) = 0\]
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Chapter 3: Functions - Exercise 3.2 [Page 11]

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RD Sharma Mathematics [English] Class 11
Chapter 3 Functions
Exercise 3.2 | Q 7 | Page 11

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