If $h(x)$ is a function whose domain is $[-8,8]$, and $g(x)=h\left(\frac x2\right)$, then the domain of $g(x)$ is an interval of what width?
I need to find the width of the domain of $g(x)$, which is the difference between the largest and smallest possible values of $x$ that make $g(x)$ defined. Since $g(x)$ is defined in terms of $h\left(\frac x2\right)$, I need to consider how the values of $\frac x2$ relate to the domain of $h(x)$. The domain of $h(x)$ is $[-8,8]$, which means that $h(x)$ is only defined when $x$ is between $-8$ and $8$, inclusive. Therefore, $g(x)$ is only defined when $\frac x2$ is between $-8$ and $8$, inclusive. I can write this as an inequality: $-8\leq \frac x2\leq 8$. To find the largest and smallest values of $x$ that satisfy this inequality, I can multiply everything by $2$. This gives me $-16\leq x\leq 16$, which is the domain of $g(x)$. The width of this interval is the difference between $16$ and $-16$, which is $32$. 