Answer
$a)$ $(f+g)(x)=4x^{3}-6x$
$b)$ $(f-g)(x)=4x^{3}+6x$
$c)$ $(f\cdot g)(x)=-24x^{4}$
$d)$ $\Big(\dfrac{f}{g}\Big)(x)=-\dfrac{2x^{2}}{3}$
Work Step by Step
$f(x)=4x^{3};$ $g(x)=-6x$
For each case, replace $f(x)$ by $4x^{3}$ and $g(x)$ by $-6x$ and simplify if possible:
$a)$ $(f+g)(x)$
$f(x)+g(x)=(4x^{3})+(-6x)=4x^{3}-6x$
$b)$ $(f-g)(x)$
$f(x)-g(x)=(4x^{3})-(-6x)=4x^{3}+6x$
$c)$ $(f\cdot g)(x)$
$f(x)\cdot g(x)=(4x^{3})(-6x)=-24x^{4}$
$d)$ $\Big(\dfrac{f}{g}\Big)(x)$
$\dfrac{f(x)}{g(x)}=\dfrac{4x^{3}}{-6x}=-\dfrac{2x^{2}}{3}$
