Chapter 2.2, Problem 4QR

To determine

To find: The function $f^{-1}$ for $f\left(x\right)={\cot }^{-1}x$ and also graph the function $f$ , $f^{-1}$ and $y=x$ in the same viewing window.

Answer to Problem 4QR

The function $f^{-1}$ for $f\left(x\right)={\cot }^{-1}x$ is given as $f^{-1}\left(x\right)=\cot x$ and the graph of function $f$ , $f^{-1}$ and $y=x$ are shown in Figure (1).

Explanation of Solution

Given information: The function is $f\left(x\right)={\cot }^{-1}x$.

Calculation:

Equate the given function to $y$.

  $\begin{array}{c}y={\cot }^{-1}x\\ \cot y=x\\ x=\cot y\end{array}$

Interchange $x$ and $y$ in above equation.

  $\begin{array}{c}x=\cot y\\ y=\cot x\\ f^{-1}\left(x\right)=\cot x\end{array}$

So, the inverse of the function $f\left(x\right)={\cot }^{-1}x$ is $f^{-1}\left(x\right)=\cot x$.

To graph the functions $y={\cot }^{-1}x$ , $y=\cot x$ and $y=x$ , 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 $y={\cot }^{-1}x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{2\text{nd}}\boxed{\text{TAN}}\boxed{1}\boxed{÷}\boxed{\text{X}}\boxed{)}$ . Now enter right hand side of the equation $y=\cot x$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{1}\boxed{÷}\boxed{(}\boxed{\text{TAN}}\boxed{\text{X}}\boxed{)}$ . Nowenter right hand side of the equation $y=x$ after the symbol ${\text{Y}}_3$ . Enter the keystrokes $\boxed{\text{X}}$.

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}={\cot }^{-1}\left(\text{X}\right)\\ {\text{Y}}_2=\cot \left(\text{X}\right)\\ {\text{Y}}_3=\text{X}\end{array}$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown below.

  

Figure (1)

Therefore, the function $f^{-1}$ for $f\left(x\right)={\cot }^{-1}x$ is given as $f^{-1}\left(x\right)=\cot x$ and the graph of function $f$ , $f^{-1}$ and $y=x$ are shown in Figure (1).