Chapter 2.2, Problem 2QR

To determine

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

Answer to Problem 2QR

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

Explanation of Solution

Given information:

The function is $f\left(x\right)=e^x$.

Calculation:

Equate the given function to $y$.

  $y=e^x$

Take logarithm both sides.

  $\begin{array}{c}\ln y=\ln e^x\\ \ln y=x\end{array}$

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

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

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

To graph the functions $y=e^x$ , $y=\ln x$ 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 $y=e^x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{2\text{nd}}\boxed{\text{LN}}\boxed{\text{X}}\boxed{)}$ . Now enter right hand side of the equation $y=\ln x$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{\text{LN}}\boxed{\text{X}}\boxed{)}$ . Now enter 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,

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

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

  

Figure (1)

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