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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) =
Options
(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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