Question

use the graph of $f(x)$ shown below to determine the domain of $f$ and the equations of any ve horizontal asymptotes.
state the domain of $f$ using interval notation: $(-\infty,-4)\cup(-4,-1)\cup(-1,\infty)$
state the equation(s) of the vertical asymptote(s). if there is more than one, separate them by a if there is not any vertical asymptote, enter dne.
vertical asymptotes:
state the equation of the horizontal asymptote. if there is not one, enter dne.
horizontal asymptote:

Explanation:

Step1: Identify vertical asymptotes

Vertical asymptotes occur where the function has a break (discontinuity) and the graph approaches infinity or negative infinity. From the domain \((-\infty,-4)\cup(-4, - 1)\cup(-1,\infty)\), the values that are excluded from the domain are \(x=-4\) and \(x = - 1\). So the equations of the vertical asymptotes are \(x=-4\) and \(x=-1\).

Step2: Identify horizontal asymptote

A horizontal asymptote is a horizontal line \(y = k\) that the graph of the function approaches as \(x\to\pm\infty\). Looking at the graph, as \(x\to\pm\infty\), the \(y\) - value approaches \(y=-1\).

Answer:

Vertical asymptotes: \(x=-4,x = - 1\)
Horizontal asymptote: \(y=-1\)