3. The function $f(x)=\frac{x}{1-2^{x}}-\frac{x}{2}$
A. is an even function but not an odd function
B. is an odd function but not an even function
C. is both an even function and an odd function
D. is neither an even function nor an odd function
The domain of $f(x)$ is $(-\infty, 0) \bigcup(0,+\infty)$. And $f(-x)=\frac{-x}{1-2^{-x}}+\frac{x}{2}=\frac{-x \cdot 2^{x}}{2^{x}-1}+\frac{x}{2}$ $=\frac{x \cdot 2^{x}-x+x}{1-2^{x}}+\frac{x}{2}=\frac{x}{1-2^{x}}-\frac{x}{2}=f(x)$, so $f(x)$ is an even function. Therefore, the answer is $A$.