Graphs and Functions
Quotient of two functions: Advanced

Suppose that the functions $f$ and $g$ are defined as follows.
\[
f(x)=\frac{5}{x+3} \quad g(x)=\frac{8}{x}
\]

Find $\frac{f}{g}$. Then, give its domain using an interval or union of intervals. Simplify your answers.
\[
\left(\frac{f}{g}\right)(x)=\square
\]

Domain of $\frac{f}{g}: \square$
\begin{tabular}{ccc}
$\frac{\square}{\square}$ & $\square^{\square}$ & $(\square, \square)$ \\
{$[\square, \square]$} & $\square \cup \square$ & $(\square, \square]$ \\
{$[\square, \square)$} & $\varnothing$ & $\infty$ \\
$-\infty$ & & \\
$x$ & &
\end{tabular}

Step 1 ：To find the quotient of the two functions, we divide $f(x)$ by $g(x)$: \(\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}=\frac{\frac{5}{x+3}}{\frac{8}{x}}\)

Step 2 ：To simplify this, we multiply the numerator and the denominator by $x(x+3)$ to get rid of the fractions: \(\left(\frac{f}{g}\right)(x)=\frac{5x}{8(x+3)}\)

Step 3 ：So, \(\left(\frac{f}{g}\right)(x)=\frac{5x}{8(x+3)}\)

Step 4 ：Now, let's find the domain of \(\frac{f}{g}\). The domain of a function is the set of all real numbers for which the function is defined. In this case, the function is undefined when the denominator equals zero. So, we set the denominator equal to zero and solve for $x$: \(8(x+3)=0\)

Step 5 ：Solving this equation gives $x=-3$. So, the function is undefined at $x=-3$. Also, the function $g(x)$ is undefined at $x=0$. Therefore, the domain of \(\frac{f}{g}\) is all real numbers except $-3$ and $0$.

Step 6 ：In interval notation, this is \(-\infty, -3) \cup (-3, 0) \cup (0, \infty)\)

Step 7 ：So, the domain of \(\frac{f}{g}\) is \(\boxed{(-\infty, -3) \cup (-3, 0) \cup (0, \infty)}\)