Example 1 Given $y=f(x)$ is a monotonically increasing odd function, its domain is $[-1,1]$, find the domain and range of the function $g(x)=$ $\sqrt{f\left(x^{2}-3\right)+f(x+1)}$.
The domain of $g(x)$ is constrained by the domain of $f(x)$. It satisfies the inequalities:
$$
\left\{\begin{array}{l}
-1 \leqslant x^{2}-3 \leqslant 1 \\
-1 \leqslant x+1 \leqslant 1 \\
f\left(x^{2}-3\right)+f(x+1) \geqslant 0
\end{array}\right.
$$

From (1), we get: $\sqrt{2} \leqslant x \leqslant 2$ or $-2 \leqslant x \leqslant-\sqrt{2}$
From (2), we get: $-2 \leqslant x \leqslant 0$
The intersection of (4) and (5) is $-2 \leqslant x \leqslant-\sqrt{2}$

Considering (3): $f\left(x^{2}-3\right) \geqslant-f(x+1)$,
Since $f(x)$ is an odd function, we have $f\left(x^{2}-3\right) \geqslant f(-x-1)$,
And since $f(x)$ is an increasing function, it follows that $x^{2}-3 \geqslant-x-1$,
Thus, $x \geqslant 1$ or $x \leqslant-2$
From (1) and (7), the domain of $g(x)$ is $\{-2\}$, and thus the range of $g(x)$ is $\{0\}$.
Analyzing this problem, it appears structurally simple but actually contains a rich array of concepts: domain, range, composite functions, odd and even properties, monotonicity, etc.

We first list the function inequalities based on the domain of $g(x)$, then, using the fact that $f(x)$ is an increasing odd function, we remove the "f" to convert it into ordinary numerical inequalities.