Question

$Iff\left(x\right)+2f\left(\frac{1}{x}\right)=3x,x\ne 0,andS=\{xϵR:f(x)=f(-x\left)\right\};thenS:$

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

Answer 1

The correct option is B

Contains exactly two elements

$f\left(x\right)+2f\left(\frac{1}{x}\right)=3x\Rightarrow f\left(\frac{1}{x}\right)+2f\left(x\right)=\frac{3}{x}\therefore 3f\left(x\right)=\frac{6}{x}-3x\therefore f\left(x\right)=(\frac{2}{x}-x)f\left(x\right)=f(-x)\Rightarrow \frac{2}{x}-x=-\frac{2}{x}+x\Rightarrow 2x-\frac{4}{x}=0\Rightarrow x=\pm \sqrt{2}$