If $f(x)$ is a function whose domain is $[-8,8]$, and $g(x)=f\left((x^2 - 2)/(x + 1)\right)$, then the domain of $g(x)$ is an interval of what width?

Answer 1

Recall that the domain of a function consists of all the values of $x$ for which the function is defined. In this case, $g(x)$ is defined only when $f\left(\frac{x^2 - 2}{x + 1}\right)$ is defined. Thus we need to consider when $\frac{x^2 - 2}{x + 1}$ is in the domain of $f(x)$. Notice that $f(x)$ is defined for all values of $x$ in the domain $[-8,8]$. Therefore, $\frac{x^2 - 2}{x + 1}$ is always defined except when $x=-1$. Thus the domain of $g(x)$ is an interval of length $\boxed{2}$ because it includes $\frac{x^2 - 2}{x + 1}$ for all $x$ except $x=-1$.