Chapter 6, Problem 5CLT

To determine

To find: The sketch for the graph of the function.

Explanation of Solution

Given:

The given function is $f\left(x\right)=\frac{1}{2}\tan \left(x-\frac{\pi }{2}\right)$ .

Calculation:

Consider the function $f\left(x\right)=\frac{1}{2}\tan \left(x-\frac{\pi }{2}\right)$ then the solution for $x$ is,

  $\begin{array}{l}x=\frac{1}{2}\tan \left(x-\frac{\pi }{2}\right)\\ x-\frac{\pi }{2}=-\frac{\pi }{2}\\ x=0\end{array}$

This shows that the function is undefined as $x=0$ .

The table for the value of the function for various values of $x$ is shown in Table 1

Table 1

$x$	$-\frac{\pi }{2}$	$-\frac{\pi }{4}$	$0$	$\frac{\pi }{4}$	$\frac{\pi }{2}$
$f\left(x\right)=\frac{1}{2}\tan \left(x-\frac{\pi }{2}\right)$	$0$	$0.5$	undefined	$-0.5$	$0$

The graph for the function is shown in Figure 1

  

Figure 1

To plot the graph with the graphing utility. Click on $Y=$ button and the then enter the equation as shown in Figure 2

  

Figure 2

Then select the window button and then select the maximum and the minimum values of the labels as shown in Figure 3.

  

Figure 3

Then select the plot button to plot the graph.

The required plot is shown in Figure 4.

  

Figure 4