Chapter 13.8, Problem 8LC

Solution

To graph: The given functions and find their similarities and differences.

The graph and range are similar but domain and vertical asymptote are different.

Given:

The functions $y=\sec x$ and $y=\csc x$ .

Calculation:

The graph of $y=\sec x$ and $y=\csc x$ is given below in the same viewing window:

  

Here red curve represents the $y=\sec x$ and the blue curve represents the $y=\csc x$ .

The graph of both the functions are similar because they have same shape that can be observed in the above given figure. The range of both the functions is $\left(-\infty ,-1\right]\cup \left[1,\infty \right)$ . However, the domain of $y=\csc x$ and $y=\sec x$ are different and due to this their vertical asymptote are also different.

Yes, the $y=\csc x$ can be transformation of $y=\sec x$ . And it can be done by shifting $\sec x$ to $\frac{\pi }{2}$ unit right.

  $\csc x=\sec \left(x-\frac{\pi }{2}\right)$

Conclusion:

Hence, the graph and range are similar but domain and vertical asymptote are different.