Example 7 The range of the function $y=\frac{x^{2}-1}{x^{2}+2}$ is $\qquad$ .
From the given, we have $x^{2}=\frac{2 y+1}{1-y} \geqslant 0$, solving this yields $-\frac{1}{2} \leqslant y<1$, therefore, the range of the function $y=\frac{x^{2}-1}{x^{2}+2}$ is $\left[-\frac{1}{2}, 1\right)$.

Solution Guidance Construct an inequality that the dependent variable satisfies, and obtain the range of the function by solving the inequality. This is another important method for studying the range of a function, or rather, this is an application of the "construction method" in solving problems related to the range of a function.