Chapter 3.4, Problem 41E

(a)

To determine

To graph:The function $d^{-1}\left(x\right)$ ..

Answer to Problem 41E

The graph of $d^{-1}\left(x\right)$ is shown in figure (1).

Explanation of Solution

Consider the given function.

  $d\left(x\right)=\left|x-4\right|$

So,

  $\begin{array}{c}dx=x-4\\ x=dx+4\\ d^{-1}\left(x\right)=x+4\end{array}$

Or,

  $\begin{array}{c}dx=-\left(x-4\right)\\ -d\left(x\right)=x-4\\ 4-d\left(x\right)=x\\ d^{-1}\left(x\right)=4-x\end{array}$

The graph of $d^{-1}\left(x\right)$ is shown below.

  

Figure (1)

(b)

To determine

To find: Whether $d^{-1}\left(x\right)$ is a function.

Answer to Problem 41E

No, $d^{-1}\left(x\right)$ is not a function.

Explanation of Solution

No, the function $d^{-1}\left(x\right)$ is not a function as it fails the vertical line test as shown in figure (1).

Therefore, the function $d^{-1}\left(x\right)$ is not a function.

(c)

To determine

To describe: The function $d^{-1}\left(x\right)$ it represent.

Answer to Problem 41E

The function $d^{-1}\left(x\right)$ it represent is stated below.

Explanation of Solution

The $f^{-1}\left(x\right)$ is a reflection of $f\left(x\right)$ over the line $y=x$ .

So it can observed that the initial graph of $f\left(x\right)$ fails the horizontal line test, so its inverse would not be a function.