Step 1 :Define the functions $f, g, h$ as $f(x)=|12 x-13|$, $g(x)=\sqrt{13+5 x}$, and $h(x)=\frac{x^{3}+2}{x^{3}}$ respectively.
Step 2 :Substitute $x=-\frac{3}{4}$ into $f(x)$ to get $f\left(-\frac{3}{4}\right)=|12(-\frac{3}{4})-13|=22$.
Step 3 :Substitute $x=2$ into $g(x)$ to get $g(2)=\sqrt{13+5(2)}=\sqrt{23}$.
Step 4 :Substitute $x=-4$ into $h(x)$ to get $h(-4)=\frac{(-4)^{3}+2}{(-4)^{3}}=\frac{31}{32}$.
Step 5 :So, the final answers are $f\left(-\frac{3}{4}\right)=\boxed{22}$, $g(2)=\boxed{\sqrt{23}}$, and $h(-4)=\boxed{\frac{31}{32}}$.