Doubt about domain of a function and how to evaluate absolute value function

Question

Suppose we have a function $f(x)$ whose domain is $|x| < 1$ and say we try to find the domain of the function $f(g(x))$ where $g(x)$ is just a simple polynomial. To make things even simpler, let us assume it is $g(x) = -\frac{x^2}{3}$. So if we were now to find the domain for $f(g(x))$ which in this case would be $f(-\frac{x^2}{3})$ then would we just need to evaluate and simplify $|-\frac{x^2}{3}| < 1$? If so then how does one go about doing that?

Answer 1

You simply do\begin{align}\left|-\frac{x^2}3\right|<1&\iff x^2<3\text{ (since $a\leqslant0\iff|a|=-a$)}\\&\iff|x|<\sqrt3.\end{align}