If 2f (x) − 3f
Question:

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

(a) $-\frac{7}{4}$

(b) $\frac{5}{2}$

(c) −1

(d) None of these

Solution:

(a) $-\frac{7}{4}$

$2 f(x)-3 f\left(\frac{1}{x}\right)=x^{2} \quad(x \neq 0)$

Replacing $x$ by $\frac{1}{x}$ :

$2 f\left(\frac{1}{x}\right)-3 f(x)=\frac{1}{x^{2}}$    ….(2)

Solving equations $(1) \&(2)$

$-5 f(x)=\frac{3}{x^{2}}+2 x^{2}$

$\Rightarrow f(x)=\frac{-1}{5}\left(\frac{3}{x^{2}}+2 x^{2}\right)$

Thus, $f(2)=\frac{-1}{5}\left(\frac{3}{4}+2 \times 4\right)$

$=\frac{-1}{5}\left(\frac{3+32}{4}\right)$

$=\frac{-7}{4}$

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