Answer
$a)$ $(f+g)(x)=5x^{2}-3x$
$b)$ $(f-g)(x)=-5x^{2}-3x$
$c)$ $(f\cdot g)(x)=-15x^{3}$
$d)$ $\Big(\dfrac{f}{g}\Big)(x)=-\dfrac{3}{5x}$, where $x\ne0$
Work Step by Step
$f(x)=-3x;$ $g(x)=5x^{2}$
For each case, replace $f(x)$ by $-3x$ and $g(x)$ by $5x^{2}$ and simplify if possible:
$a)$ $(f+g)(x)$
$f(x)+g(x)=(-3x)+(5x^{2})=5x^{2}-3x$
$b)$ $(f-g)(x)$
$f(x)-g(x)=(-3x)-(5x^{2})=-5x^{2}-3x$
$c)$ $(f\cdot g)(x)$
$f(x)\cdot g(x)=(-3x)(5x^{2})=-15x^{3}$
$d)$ $\Big(\dfrac{f}{g}\Big)(x)$
$\dfrac{f(x)}{g(x)}=\dfrac{-3x}{5x^{2}}=\dfrac{-3}{5x}=-\dfrac{3}{5x}$, where $x\ne0$
