Question

If $f(x)+2f\left(\frac{1}{x}\right)=3x,x\ne 0$, and $S=\{x\in R:f(x)=f(-x)\}$; then $S$ :

• A. is an empty set.
• B. contains exactly one element.
• C. contains exactly two elements.
• D. contains more than two elements.

Answer 1

Answer: (C)

$f\left(x\right)+2f\left(\frac{1}{x}\right)=3x$
Replace x by $\frac{1}{x},f\left(\frac{1}{x}\right)+2f\left(x\right)=\frac{3}{x}$
$\Rightarrow \frac{3x-f\left(x\right)}{2}=\frac{\frac{3}{x}-2f\left(x\right)}{1}$
$\Rightarrow 3x-f\left(x\right)=\frac{6}{x}-4f\left(x\right)$
$\Rightarrow f\left(x\right)=\frac{2}{x}-x$
$f\left(x\right)=f\left(-x\right)$
$\Rightarrow \frac{2}{x}-x=-\frac{2}{x}+x$
$\Rightarrow \frac{4}{x}=2x$
$\Rightarrow x=\pm \sqrt{2}$