5. (11th "Hope Cup" Invitational Competition Question) The function $f(x)=\log _{\frac{1}{3}}\left(2 x^{2}+2 x \sqrt{x^{2}+1}+1\right)^{x}$ is $(\quad$ ).
A. Even function
B. Odd function
C. Both odd and even function
D. Neither odd nor even function
5. A. Reason: It is easy to see that $f(x)=\log _{\frac{1}{3}}\left(x+\sqrt{x^{2}+1}\right)^{2 x}$, thus $f(-x)=\log _{\frac{1}{3}}\left(-x+\sqrt{x^{2}+1}\right)^{-2 x}=\log _{\frac{1}{3}}\left(\frac{1}{-x+\sqrt{x^{2}+1}}\right)^{2 x}=\log _{\frac{1}{3}}\left(x+\sqrt{x^{2}+1}\right)^{2 x}$, hence $f(x)=f(-x)(x \in \mathbf{R})$, so $f(x)$ is an even function.