Chapter 1.6, Problem 26E

To determine

To show: The function $y=\tan x$ is one-one and sketch graph of its inverse for $-\frac{\pi }{2}<x<\frac{\pi }{2}$ .

Explanation of Solution

Given information:

The given function is $y=\tan x$ and the interval is $-\frac{\pi }{2}<x<\frac{\pi }{2}$ .

Proof:

To graph a function $y=\tan x$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=\tan x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{TAN}}\boxed{\text{X}}\boxed{)}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\tan \left(\text{X}\right)$

Now, press the $\boxed{\text{WINDOW}}$ key and adjust the value of $x$ to $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ . Press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of both the equations as shown below.

  

Figure (1)

As observed from the graph of function, the function $y=\tan x$ doesn’t repeat its value for $-\frac{\pi }{2}<x<\frac{\pi }{2}$ . So, the given function $y=\tan x$ is one-one.

The inverse of the function $y=\tan x$ is the function $f\left(x\right)={\tan }^{-1}\left(x\right)$ .

Now graph the inverse function.

To graph a function $f\left(x\right)={\tan }^{-1}\left(x\right)$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y={\tan }^{-1}\left(x\right)$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{2nd}}\boxed{\text{TAN}}\boxed{\text{X}}\boxed{)}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}={\tan }^{-1}\left(\text{X}\right)$

Now, press the $\boxed{\text{WINDOW}}$ key and adjust the value of $x$ to $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ . Press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of both the equations as shown below.

  

Figure (2)

Hence, the given function is one-one and graph of its inverse is shown in Figure (2).