1. Find all functions $f: \mathbf{Q} \rightarrow \mathbf{R} \backslash\{0\}$, such that for any rational numbers $x, y$, we have
$$
\begin{array}{l}
f^{2}(x) f(2 y)+f^{2}(y) f(2 x) \\
=2 f(x) f(y) f(x+y) .
\end{array}
$$
$(2015$, Taiwan Mathematical Olympiad Training Camp)
Let $g(x)=\frac{f(x)}{f(0)}$, then $g(x+y)=g(x) g(y)$.
First, solve according to the variant of Cauchy's functional equation, then further derive $f(x)=b c^{x}(b \neq 0, c>0)$.