1. Given that the domain of the function $f(x)$ is $[-1,1]$, find the domain of $f(a x)+f\left(\frac{x}{a}\right)$ (where $a>0$).
1. From the problem, we know that the range of $x$ is $-\frac{1}{a} \leqslant x \leqslant \frac{1}{a}$, or $-a \leqslant x \leqslant a$. When $a \geqslant 1$, the solution is $-\frac{1}{a} \leqslant x \leqslant \frac{1}{a}$; when $0<a<1$, the solution is $-a \leqslant x \leqslant a$. Therefore, for any given $a \in(0,1)$, the domain of the function $f(a x)+f\left(\frac{x}{a}\right)$ is $[-a$, $a]$; for any given $a \in[1,+\infty)$, the domain of this function is $\left[-\frac{1}{a}, \frac{1}{a}\right]$.