4. If $a>0, a \neq 1, F(x)$ is an odd function, then $G(x)=F(x)\left(\frac{1}{a^{x}-1}+\frac{1}{2}\right)$ is
A. Odd function
B. Even function
C. Neither odd nor even function
D. The odd or even nature depends on $a$
$\begin{array}{l}\text { 4. B } G(-x)=F(-x)\left(\frac{1}{a^{-x}-1}+\frac{1}{2}\right)=-F(x)\left(\frac{a^{x}}{1-a^{x}}+\frac{1}{2}\right)=-F(x) \frac{2 a^{x}+1-a^{x}}{2\left(1-a^{x}\right)}= \\ -F(x) \frac{a^{x}+1}{2\left(1-a^{x}\right)}=F(x) \frac{a^{x}+1}{2\left(a^{x}-1\right)}=F(x)\left(\frac{1}{a^{x}-1}+\frac{1}{2}\right)=G(x)\end{array}$