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If 3 F ( X ) + 5 F ( 1 X ) = 1 X − 3 for All Non-zero X, Then F(X) =(A)1 14 ( 3 X + 5 X − 6 )(B) 1 14 ( − 3 X + 5 X − 6 )(C) 1 14 ( − 3 X + 5 X + 6 )(D) None of These - CBSE (Science) Class 11 - Mathematics

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Question

If \[3f\left( x \right) + 5f\left( \frac{1}{x} \right) = \frac{1}{x} - 3\]  for all non-zero x, then f(x) =

  • (a)  \[\frac{1}{14}\left( \frac{3}{x} + 5x - 6 \right)\]

  • (b)  \[\frac{1}{14}\left( - \frac{3}{x} + 5x - 6 \right)\]

  • (c) \[\frac{1}{14}\left( - \frac{3}{x} + 5x + 6 \right)\]

  • (d) None of these

     

Solution

(d) None of these

\[3f\left( x \right) + 5f\left( \frac{1}{x} \right) = \frac{1}{x} - 3\]

\[\text{ Multiplying (1) by } 3: \]

\[15 f\left( \frac{1}{x} \right) + 9 f(x) = \frac{3}{x} - 9 . . . . . (2)\]

\[\text{ Replacing x by}  \frac{1}{x}\text{ in } (1): \]

\[3 f\left( \frac{1}{x} \right) + 5 f(x) = x - 3 \]

\[\text{ Multiplying by } 5: \]

\[15 f\left( \frac{1}{x} \right) + 25 f(x) = 5x - 15 . . . . (3)\]

\[\text{ Solving (2) and (3) } : \]

\[ - 16 f(x) = \frac{3}{x} - 5x + 6\]

\[ \Rightarrow f(x) = \frac{1}{16}\left( - \frac{3}{x} + 5x - 6 \right)\]

 

Notes

Disclaimer: The question in the book has some error, so, none of the options are matching with the solution. The solution is created according to the question given in the book.

 
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Solution If 3 F ( X ) + 5 F ( 1 X ) = 1 X − 3 for All Non-zero X, Then F(X) =(A)1 14 ( 3 X + 5 X − 6 )(B) 1 14 ( − 3 X + 5 X − 6 )(C) 1 14 ( − 3 X + 5 X + 6 )(D) None of These Concept: Functions.
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