Question

The function $f$ is not defined for $x=0$, but for all non zero real numbers $x$, $f(x)+2f\left(\frac{1}{x}\right)=3x$. The equation $f(x)=f(-x)$ is satisfied by

• A. exactly one real number
• B. exactly two real numbers
• C. no real numbers
• D. all non zero real numbers.

Answer 1

Answer: (B)

$f(x)+2f\left(\frac{1}{x}\right)=3x$....(1)
$x\to \frac{1}{x},2f(x)+f\left(\frac{1}{x}\right)=\frac{3}{x}$....(2)
or $4f(x)+2f\left(\frac{1}{x}\right)=\frac{6}{x}$(3)
(3) -(1)
$3f(x)=\frac{6}{x}-3x$

now $f(x)=f(-x)$
$\frac{2}{x}-x=-\frac{2}{x}+x\Rightarrow \frac{4}{x}=2x$
$\Rightarrow x^2=2\Rightarrow x=\sqrt{2}\text{ or }-\sqrt{2}\Rightarrow (B)$