7・110 Among the four functions listed below, the function $f(x)$ that satisfies the relation
$$
2 f(x)-f\left(\frac{1}{x}\right)=\frac{1}{x}
$$

is
(A) $\frac{x}{3}+\frac{2}{3 x}$.
(B) $\frac{3}{x}+\frac{3}{2} x$.
(C) $\frac{2}{3} x+\frac{1}{3 x}$.
(D) $\frac{3}{2} x+\frac{1}{3 x}$.
(China Beijing High School Mathematics Competition, 1987)
[Solution] Let $y=\frac{1}{x}$, we get $2 f\left(\frac{1}{y}\right)-f(y)=y$.
Substituting $y$ with $x$ gives $\quad-f(x)+2 f\left(\frac{1}{x}\right)=x$.
Multiplying the original equation by 2 gives $\quad 4 f(x)-2 f\left(\frac{1}{x}\right)=\frac{2}{x}$.
(1)
(2) yields
$$
3 f(x)=\frac{2}{x}+x \text {. }
$$

Therefore,
$$
f(x)=\frac{2}{3 x}+\frac{x}{3} \text {. }
$$

Hence, the answer is $(A)$.