If $3 f(x)+5 f\left(\frac{1}{x}\right)=\frac{1}{x}-3$ for all non-zero $x$, then $f(x)=$
(a) $\frac{1}{14}\left(\frac{3}{x}+5 x-6\right)$
(b) $\frac{1}{14}\left(-\frac{3}{x}+5 x-6\right)$
(c) $\frac{1}{14}\left(-\frac{3}{x}+5 x+6\right)$
(d) None of these
(d) None of these
$3 f(x)+5 f\left(\frac{1}{x}\right)=\frac{1}{x}-3$ ...91)
Multiplying (1) by 3 :
15 f\left(\frac{1}{x}\right)+9 f(x)=\frac{3}{x}-9 \ldots \ldots(2)
Replacing $x$ by $\frac{1}{x}$ in $(1)$ :
$3 f\left(\frac{1}{x}\right)+5 f(x)=x-3$
Multiplying by 5 :
$15 f\left(\frac{1}{x}\right)+25 f(x)=5 x-15 \ldots(3)$
Solving $(2)$ and $(3)$ :
$-16 f(x)=\frac{3}{\mathrm{x}}-5 \mathrm{x}+6$
$\Rightarrow f(x)=\frac{1}{16}\left(-\frac{3}{\mathrm{x}}+5 \mathrm{x}-6\right)$
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.