Chapter 1, Problem 53RE

(a)

To determine

To find:The inverse of the given function and also verify that $\left(f\circ f^{-1}\right)\left(x\right)=\left(f^{-1}\circ f\right)\left(x\right)=x$ .

Answer to Problem 53RE

Theinverse of the given function is $f^{-1}\left(x\right)=\frac{2-x}{3}$ and it is verified that $\left(f\circ f^{-1}\right)\left(x\right)=\left(f^{-1}\circ f\right)\left(x\right)=x$ .

Explanation of Solution

Given information:The function is $f\left(x\right)=2-3x$ .

Calculation:

Substitute $y$ for $f\left(x\right)$ in the given function and solve the equation for $x$ in terms of $y$ .

  $\begin{array}{c}y=2-3x\\ 3x=2-y\\ x=\frac{2-y}{3}\end{array}$

Interchange $x$ and $y$ , and substitute $f^{-1}\left(x\right)$ for $y$ .

  $\begin{array}{c}y=\frac{2-x}{3}\\ f^{-1}\left(x\right)=\frac{2-x}{3}\end{array}$

Verify:

The composite function $\left(f\circ f^{-1}\right)\left(x\right)$ can be written as:

  $\left(f\circ f^{-1}\right)\left(x\right)=f\left(f^{-1}\left(x\right)\right)$

Substitute $\frac{2-x}{3}$ for $f^{-1}\left(x\right)$ to find $\left(f\circ f^{-1}\right)\left(x\right)$ .

  $\begin{array}{c}\left(f\circ f^{-1}\right)\left(x\right)=f\left(\frac{2-x}{3}\right)\\ =2-3\left(\frac{2-x}{3}\right)\\ =2-2+x\\ =x\end{array}$

The composite function $\left(f^{-1}\circ f\right)\left(x\right)$ can be written as:

  $\left(f^{-1}\circ f\right)\left(x\right)=f^{-1}\left(f\left(x\right)\right)$

Substitute $2-3x$ for $f\left(x\right)$ to find $\left(f^{-1}\circ f\right)\left(x\right)$ .

  $\begin{array}{c}\left(f^{-1}\circ f\right)\left(x\right)=f^{-1}\left(2-3x\right)\\ =\frac{2-\left(2-3x\right)}{3}\\ =\frac{3x}{3}\\ =x\end{array}$

It is verified that $\left(f\circ f^{-1}\right)\left(x\right)=\left(f^{-1}\circ f\right)\left(x\right)=x$ .

Therefore, the inverse of the given function is $f^{-1}\left(x\right)=\frac{2-x}{3}$ and it is verified that $\left(f\circ f^{-1}\right)\left(x\right)=\left(f^{-1}\circ f\right)\left(x\right)=x$ .

(b)

To determine

To graph:The function $f\left(x\right)=2-3x$ and its inverse in the same viewing window.

Explanation of Solution

Given information:The function is $f\left(x\right)=2-3x$ .

Graph:

From part (a), the inverse of the given function is $f^{-1}\left(x\right)=\frac{2-x}{3}$ .

To plot the graph of $f$ and $f^{-1}$ in the same viewing window, use the following steps of graphing calculator.

First press “ON” button on graphical calculator and press $\boxed{\text{Y}=}$ key. Write the right hand side of the function $f\left(x\right)=2-3x$ after ${\text{Y}}_{\text{1}}$ . Enter the keystrokes $\boxed{\text{2}}\boxed{-}\boxed{3}\boxed{\text{X}}$ .

Press $\boxed{\text{ENTER}}$ and write the eight hand side of the function $f^{-1}\left(x\right)=\frac{2-x}{3}$ after ${\text{Y}}_{\text{2}}$ . Enter the keystrokes $\boxed{(}\boxed{\text{2}}\boxed{-}\boxed{\text{X}}\boxed{)}\boxed{÷}\boxed{3}$ . Adjust the window at $\left[-6,6\right]$ by $\left[-4,4\right]$ .

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 $y=x$ .