Chapter 4.7, Problem 71E

Solution

(a)

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

Given information:

  

Graph:

  

Interpretation: the graph has the horizontal asymptote which is $y=\frac{\pi }{2}$

(b)

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

Given information:

  

Graph:

  

Interpretation: the graph has the horizontal asymptote which is $y=\frac{\pi }{2}$ and $y=\frac{3\pi }{2}$

(c)

To graph: what is the graph related to function.

Given information: we are given the function:

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

Graph:

  

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

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

  $\begin{array}{l}x\to -\infty \Rightarrow \frac{1}{x}\to 0\Rightarrow g(x)\to \frac{\pi }{2}\\ x\to \infty \Rightarrow \frac{1}{x}\to 0\Rightarrow g(x)\to \frac{\pi }{2}\end{array}$

Horizontal asymptote: $y=\frac{\pi }{2}$ .

(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 decreasing on $(-\infty ,1)$ and increasing on $(1,\infty )$ .

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