Single Variable Calculus, Chapter 1, 1.3, Section 1.3, Problem 29

Find the functions $f+g, f-g, f \cdot g,$ and $f/g$ and their domains

$f(x) = x^3 + 2x^2, \qquad g(x) = 3x^2-1$


$
\begin{equation}
\begin{aligned}

@ f+g\\
f+g =& f(x)+g(x) && \text{ Substitute the given values of the function $f(x)$ and $g(x)$}\\
f + g=& x^3+2x^2+3x^2-1 && \text{ Combine like terms }
\end{aligned}
\end{equation}
$


$\boxed{f + g = x^3+5x^2-1 } $

$\boxed{\text{ The domain of this function is :} (-\infty,\infty)}$



$
\begin{equation}
\begin{aligned}
@f-g\\
f-g =& f(x)-g(x) && \text{ Substitute the given values of the function $f(x)$ and $g(x)$}\\
f-g =& x^3+2x^2-(3x^2-1) && \text{ Simplify the equation}\\
f-g =& x^3+2x^2-3x^2+1 && \text{ Combine like terms}
\end{aligned}
\end{equation}
$


$\boxed{f-g = x^3-x^2+1}$

$\boxed{\text{ The domain of this function is :} (-\infty,\infty)}$



$
\begin{equation}
\begin{aligned}
@f \cdot g\\
f \cdot g =& f(x).g(x) && \text{ Substitute the given values of the function $f(x)$ and $g(x)$}\\

f \cdot g =& (x^3+2x^2)(3x^2-1) && \text{ Using FOIL method}
\end{aligned}
\end{equation}
$


$\boxed{ f \cdot g = 3x^5 + 6x^4 - x^3 - 2x^2}$
$\boxed{\text{ The domain of this function is :} (-\infty,\infty)}$


$
\begin{equation}
\begin{aligned}

@f/g\\
f/g =& f(x)/g(x) && \text{ Substitute the given values of the function $f(x)$ and $g(x)$}\\



\end{aligned}
\end{equation}
$


$\displaystyle f/g = \frac{x^3+2x^2}{3x^2-1}$
$\boxed{\text{ The domain of this function is :} (-\infty, -\sqrt{\frac{1}{3}}) \bigcup(-\sqrt{\frac{1}{3}},\sqrt{\frac{1}{3}}) \bigcup (\sqrt{\frac{1}{3}},\infty)}$