Given the functions:
[tex]\begin{gathered} g(x)=x^2-64 \\ h(x)=\sqrt[]{8x+512} \end{gathered}[/tex]1. (h/g)(x)
[tex](\frac{h}{g})(x)=\frac{h(x)}{g(x)},\text{ g(x)}\ne0[/tex]We substitute the functions and solve:
[tex](\frac{h}{g})(x)=\frac{\sqrt[]{8x+512}}{x^2-64}[/tex]Factor the common term 8:
[tex]=\frac{\sqrt[]{8(x+64)}}{x^2-64}[/tex]And we have:
[tex]\begin{gathered} x^2-64\ne0 \\ x^2-64+64\ne0+64 \\ x^2\ne64 \\ x\ne8\text{ and x}\ne-8 \end{gathered}[/tex]