Question

question 5
what is the vertical asymptote and range of the following function?
o no vertical asymptotes and range $(-\infty,\infty)$
o va $y=-2$ and range $(-\infty,-2)(-2,\infty)$
o va $y=-3$ and range $(-\infty,-3)(-3,\infty)$
o va $y=0$ and range $(-\infty,0)(0,\infty)$

Explanation:

Step1: Recall the definition of vertical asymptote

A vertical asymptote is a vertical line \(x = a\) where the function approaches infinity or negative infinity as \(x\) approaches \(a\). From the graph, we can see that the function has a vertical asymptote at \(x=-2\). But wait, no, looking at the options, we need to check the range. The range is the set of all \(y -\)values of the function.

Step2: Analyze the horizontal behavior (for range)

As \(x\) approaches positive or negative infinity, the function approaches \(y = - 2\). The function never touches \(y=-2\). For the range, we consider the set of \(y\) values. The function has two parts. As \(x\to-\infty\) (left - hand side of the vertical asymptote), \(y\) approaches \(-2\) from below (\(y<-2\)), and as \(x\) approaches the vertical asymptote from the right (\(x\) values just greater than the vertical asymptote), \(y\) approaches \(-\infty\). So the range is \((-\infty,-2)\)

Answer:

O VA \(y=-2\) and Range \((-\infty,-2)\)