Answer
Domain: All real numbers except $x= -4, \ x= 4$
Vertical asymptote: $x=4%$
For $x= -4$ there is a hole because in the graph
Work Step by Step
Given \begin{equation}
g(x)=\frac{3 x+12}{x^2-16}.
\end{equation} The domain of a rational function consists of all real numbers except numbers that makes the denominator equal to zero. We need to set the expression of the denominator to zero to find the numbers that make the denominator zero and exclude them from the domain of the function.
\begin{equation}
\begin{aligned}
x^2-16&=0 \\
x&= \pm\sqrt{16}\\
x&= \pm 4.
\end{aligned}
\end{equation} The function can be rewritten as: \begin{equation}
f(x)=\frac{3 (x+4)}{(x+4)(x-4)}.
\end{equation} Domain: All real numbers except $x= -4, \ x= 4$
$x =4 $ represents a vertical asymptote because the factor $(x-4)$ does not simplify with the numerator. The value of $x= -4$ represents the $x$-coordinate of a hole because the factor $(x+4)$ does simplify with the numerator. Use your calculator to check the graph.
