Question

If $2f\left(x\right)-3f\left(\frac{1}{x}\right)=x^2(x\ne 0)$, then f(2) is equal to

• A. $-\frac{7}{4}$
• B. $\frac{5}{2}$
• C. $-1$
• D. None of these

Answer 1

The correct option is A

$-\frac{7}{4}$

$2f\left(x\right)-3f\left(\frac{1}{x}\right)=x^2(x\ne 0)\dots \dots \left(1\right)$
Replacing x by $\frac{1}{x}$:
$2f\left(\frac{1}{x}\right)-3f\left(x\right)=\frac{1}{x^2}\dots \dots \left(2\right)$
Solving equations (1) and (2)
$-5f\left(x\right)=\frac{3}{x^2}+2x^2\Rightarrow f\left(x\right)=\frac{-1}{5}(\frac{3}{x^2}+2x^2)$
Thus, $f\left(2\right)=\frac{-1}{5}(\frac{3}{4}+2\times 4)$
$=\frac{-1}{5}\left(\frac{3+32}{4}\right)$
$=\frac{-7}{4}$