Chapter 0.5, Problem 27E

To determine

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

Explanation of Solution

Given information:

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

Graph:

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

Let t be the parameter for the function $f\left(x\right)$ , that is. $x=t$ , then

Parametric equation for $f\left(x\right)$ :

  $x=t,y=2^{-t}$ … (1)

Parametric equation for $f^{-1}\left(x\right)$ :

Let $y=f\left(x\right)$, then $f^{-1}\left(y\right)=x$.

  $\begin{array}{c}y=2^{-x}\\ \ln \left(y\right)=\ln \left(2^{-x}\right)\\ =-x\ln 2\\ x=-\frac{\ln y}{\ln 2}\\ f^{-1}\left(y\right)=-\frac{\ln y}{\ln 2}\because f^{-1}\left(y\right)=x\end{array}$

Choose parameter $y=t$ then $f^{-1}\left(t\right)=-\ln \left(t\right)/\ln 2$ , therefore,

  $x=t,y=-\ln \left(t\right)/\ln 2$ … (2)

Parametric equation for $y=x$:

Choose parameter $x=t$ , then

  $x=t,y=t$ .

The graphs of $f,f^{-1},y=t$ are shown in Fig. (1).

  

Interpretation:

The graph of the curves $f,f^{-1},y=x$ coincide at point $\left(0.641,0.641\right)$ .