Chapter 4.7, Problem 72E

Solution

(a)

To graph: what is in the left one side graph.

Given information:

  

Graph:

  

Interpretation: the graph has the horizontal asymptote which is $y=0$

(b)

To graph: what is in the right one side graph.

Given information:

  

Graph:

  

Interpretation: the graph has the horizontal asymptote which is $y=0$ and $y=\pi$

(c)

To graph: what is the graph related to function.

Given information: we are given the function:

  $\begin{array}{l}f(x)={\sin }^{-1}x\\ g(x)={\sin }^{-1}\left(\frac{1}{x}\right)\end{array}$

Graph:

  

Therefore, the graph describing the function $g(x)$ is the one the left.

Interpretation: we determine the horizontal asymptotes of the function $g(x)$ :

  $\begin{array}{l}Whenx\to -\infty \\ then\frac{1}{x}\to 0i.e.g(x)\to 0\\ Andwhenx\to \infty \\ then\frac{1}{x}\to 0i.e.g(x)\to 0\end{array}$

Horizontal asymptote: $y=0$ .

(d)

To graph: increasing in both intervals.

Given information: we are given the function

  

Graph:

  

Therefore the graph describing the function is the one the left.

Interpretation:

The function described by the left graph is increasing on $(-\infty ,-1)\cup (1,\infty )$

While the function described by the right graph is increasing on $(-\infty ,1)$ and decreasing on $(1,\infty )$ .

Thus the function decreasing on both the connected intervals is the one described by the left graph.