Chapter SH, Problem 8.8EP

To determine

To find: the asymptotes, domain and range of the function.

Answer to Problem 8.8EP

The domain here is $\left(-\infty ,-2\right)\cup \left(-2,\infty \right)$ , range here will be from $\left(-\infty ,1\right)\cup \left(1,\infty \right)$ , Vertical asymptote is at $\text{x=-2}$ and Horizontal asymptote is at $\text{y=1}$ .

Explanation of Solution

Given:

  

Concept used:

The value $\text{x}$ or x-axis represent the domain of the graph.

The value of $\text{y}$ or y-axis represent the ranges of the graph.

Similarly, the graph will have horizontal asymptote at $\text{y=a}$ if the denominator is zero at $\text{y=a}$ and numerator isn’t zero at $\text{y=a}$ .

If $\text{n<m}$ the x-axis is the horizontal asymptote.

If $\text{n=m}$ then the line $\text{y=ab}$ is the horizontal asymptote.

If $\text{n>m}$ there will be no asymptotes.

Calculation:

From the graph it can easily check the asymptotes:

Vertical asymptote: $\text{x=-2}$ .

Horizontal asymptote: $\text{y=1}$ .

The graph of $\text{y}$ is the range and the graph of $\text{x}$ is domain.

Range here will be from:

  $\left(-\infty ,1\right)\cup \left(1,\infty \right)$ .

Since, at $\text{y=1}$ there is asymptote.

At $\text{x=-2}$ there is asymptote.

Similarly, domain here is:

  $\left(-\infty ,-2\right)\cup \left(-2,\infty \right)$ .

Hence, domain here is $\left(-\infty ,-2\right)\cup \left(-2,\infty \right)$ , range here will be from $\left(-\infty ,1\right)\cup \left(1,\infty \right)$ , Vertical asymptote is at $\text{x=-2}$ and Horizontal asymptote is at $\text{y=1}$ .