Chapter 1.5, Problem 27E

To determine

To graph: The functions $f$ , $f^{-1}$ and $y=x$ by using parametric graphing.

Explanation of Solution

Given information: The function is $f\left(x\right)=2^{-x}$.

Graph:

The parametric equation be $x=t$.

Substitute t for x in the given function to find parametric equations.

  $\begin{array}{l}y=2^{-x}\\ y=2^{-t}\end{array}$

The parametric equations for the function $f\left(x\right)$ form $-\infty <t<\infty$ are

  $\begin{array}{c}{x}_1=t\\ y_1=2^{-t}\end{array}$

Interchange the $x$ and $y$ in the parametric version of $f\left(x\right)$ to write the parametric equations from $-\infty <t<\infty$ of $f^{-1}\left(x\right)$ as:

  $\begin{array}{c}{x}_2=2^{-t}\\ y_2=t\end{array}$

The parametric equations of $y=x$ from $-\infty <t<\infty$ can be written as:

  $\begin{array}{c}{x}_3=t\\ y_3=t\end{array}$

To plot the graph of $f$ , $f^{-1}$ and $y=x$ , use the following steps of graphing calculator.

Step 1: First press “ON” button on graphical calculator. Press $\boxed{\text{MODE}}$ and under the Func select the Par option.

Step 2: Enter the keystrokes $\boxed{\text{T}}$ after the symbol ${\text{X}}_{\text{1T}}$ and $\boxed{\text{2}}\boxed{\wedge }\boxed{(-)}\boxed{\text{T}}$ after the symbol ${\text{Y}}_{\text{1T}}$ for the function $f\left(x\right)$.

Step 3: Enter the keystrokes $\boxed{\text{2}}\boxed{\wedge }\boxed{(-)}\boxed{\text{T}}$ after the symbol ${\text{X}}_{\text{2T}}$ and $\boxed{\text{T}}$ after the symbol ${\text{Y}}_{\text{2T}}$ for the function $f^{-1}\left(x\right)$.

Step 4: Enter the keystrokes $\boxed{\text{T}}$ after the symbol ${\text{X}}_{\text{3T}}$ and $\boxed{\text{T}}$ after the symbol ${\text{Y}}_{\text{3T}}$ for the function $y=x$.

Step 5: Set the window for T as $\left[-4.5,4.5\right]$ by $\left[-3,3\right]$.

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

  

Figure (1)

Interpretation: From the above graph it can be observed that the graph of functions $f$ and $f^{-1}$ is symmetric about the line $y=x$.