11. Let $f(x), g(x)$ be odd and even functions defined on $\mathbf{R}$, respectively, and $f(x)+g(x)=2^{x}$. If for $x \in[1,2]$, the inequality $a f(x)+g(2 x) \geqslant 0$ always holds, then the range of the real number $a$ is
Given
$f(x)=\frac{1}{2}\left(2^{x}-2^{-x}\right), g(x)=\frac{1}{2}\left(2^{x}+2^{-x}\right)$, when $x \in[1,2]$, $f(x)>0$, thus
$$
a \geqslant-\frac{g(2 x)}{f(x)}=-\frac{2^{2 x}+2^{-2 x}}{2^{x}-2^{-x}}=-\frac{\left(2^{x}-2^{-x}\right)^{2}+2}{2^{x}-2^{-x}} \xlongequal{t=2^{x}-2^{-x} \in\left[\frac{3}{2}, \frac{15}{4}\right]}-\left(t+\frac{2}{t}\right) \in\left[-\frac{257}{60},-\frac{17}{6}\right],
$$

Therefore, the range of the real number $a$ is $\left[-\frac{17}{6},+\infty\right)$.