Chapter 4.6, Problem 82E

To determine

To Find: To find the domain, intercepts, and asymptotes of the given function.

Answer to Problem 82E

Domain is $R$

Intercepts of x and y are $-1,1\text{and}\text{undefined}\text{.}$ and

Asymptotes of vertical and horizontal are infinite.

Explanation of Solution

Given information: Given $y=\ln x^4$

Calculation:

The given function is $y=\ln x^4$

For calculating domain, we have

  $\ln x^4$ which is defined for all values of x.

Thus, the domain of the given function is all real numbers that is $R.$ .

For calculating intercepts, we have

  $\begin{array}{l}\text{For}x\text{-intercept}\text{we}\text{put}y=0,\text{then}\\ \ln x^4=0\\ 4\ln x=0\\ \ln x=0\\ \therefore x=-1\text{and}1\\ \text{and}\text{for}y\text{-intercept}\text{we}\text{put}x=0,\text{then}\\ y=\ln 0^4\\ y=\text{undefined}\end{array}$

Thus, the required intercept for $x\text{and}y\text{is}-1,1\text{and}\text{undefined}\text{.}$

For calculating asymptotes, we have

Vertical asymptotes

Since when $x\text{tends}\text{to}\text{infinite}\text{we}\text{get}\underset{x\to \infty }{\lim }(\ln x^4)=\infty$

  $\therefore \text{there}\text{are}\text{infinite}\text{vertical}\text{asymptotes}$

Also, for Horizontal asymptotes, we have

  $\begin{array}{l}\because \underset{x\to \infty }{\lim }\ln x^4=\infty \\ \therefore \text{The}\text{given}\text{function}\text{have}\text{infinite}\text{horizontal}\text{asymptotes}\end{array}$

Thus, the vertical and horizontal asymptotes areinfinite.