1981 AHSME Problems/Problem 17

Problem

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

$\textbf{(A)}\ \text{exactly one real number} \qquad \textbf{(B)}\ \text{exactly two real numbers} \qquad\textbf{(C)}\ \text{no real numbers}\qquad \\ \textbf{(D)}\ \text{infinitely many, but not all, non-zero real numbers} \qquad\textbf{(E)}\ \text{all non-zero real numbers}$