Question

If for non-zero x, $2f\left(x\right)+3f\left(\frac{1}{x}\right)=\frac{1}{x}-5$, then the value of f(2) is

Answer 1

$2f\left(x\right)+3f\left(\frac{1}{x}\right)=\frac{1}{x}-5$ .....[1]
$\mathrm{Let}f\left(x\right)=p\mathrm{and}f\left(\frac{1}{x}\right)=q\left[1\right]\Rightarrow 2p+3q=\frac{1}{x}-5.....\left(2\right]\mathrm{Substituting}\frac{1}{x}\mathrm{in}\mathrm{place}\mathrm{of}x\mathrm{in}\left(1\right],\mathrm{we}\mathrm{get}:2f\left(\frac{1}{x}\right)+3f\left(x\right)=x-5\mathrm{or},3p+2q=x-5...\left(3\right]\mathrm{Solving}\left(2\right]\mathrm{and}\left(3\right]\mathrm{we}\mathrm{get}:p=-\frac{1}{5}\left(\frac{2}{x}-3x+5\right]\Rightarrow f\left(x\right)=-\frac{1}{5}\left(\frac{2}{x}-3x+5\right]\therefore f\left(2\right)=-\frac{1}{5}\left(\frac{2}{2}-3\left(2\right)+5\right]=-\frac{1}{5}\times 0=0$