(2011 Liaoning Province Liberal Arts College Entrance Examination Question 6) If the function $f(x)=\frac{x}{(2 x+1)(x-a)}$ is odd, then $a=(\quad)$.
(A) $\frac{1}{2}$
(B) $\frac{2}{3}$
(C) $\frac{3}{4}$
(D) 1
Clever solution: Since the domain of an odd function is symmetric about the origin, and the domain of the function $f(x)=\frac{x}{(2 x+1)(x-a)}$ is $x \neq-\frac{1}{2}$ and $x \neq a$, it follows that $a=\frac{1}{2}$. Therefore, the correct choice is (A).