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. $\frac{1}{14}(\frac{3}{x}+5x-6)$
• B. $\frac{1}{14}(-\frac{3}{x}+5x-6)$
• C. $\frac{1}{14}(-\frac{3}{x}+5x+6)$
• D. None of these

Answer 1

The correct option is D

None of these

$3f\left(x\right)+5f\left(\frac{1}{x}\right)=\frac{1}{x}-3\dots \dots \left(1\right)$
Multiplying (1) by 3:
$15f\left(\frac{1}{x}\right)+9f\left(x\right)=\frac{3}{x}-9\dots \dots \left(2\right)$
Replacing $x$ by $\frac{1}{x}$ in (1):
$3f\left(\frac{1}{x}\right)+5f\left(x\right)=x-3$
Multiplying by 5:
$15f\left(\frac{1}{x}\right)+25f\left(x\right)=5x-15\dots \dots \left(3\right)$
Solving (2) and (3) :
$-16f\left(x\right)=\frac{3}{x}-5x+6\Rightarrow f\left(x\right)=\frac{1}{16}(-\frac{3}{x}+5x-6)$
$\therefore$ (d) none of these