Chapter 1.9, Problem 55E

(a)

To determine

To find: inverse of the function of f .

Answer to Problem 55E

The inverse of the given function is $f^{-1}(x)=x^3+1$ .

Explanation of Solution

Given information:

The function $f(x)=\sqrt[3]{x-1}$

Calculation:

Let the function $y=f(x)=\sqrt[3]{x-1}$

  $\begin{array}{l}y=\sqrt[3]{x-1}\\ y^3=x-1\\ x=y^3+1\end{array}$

  $\begin{array}{l}{f}^{-1}(y)=y^3+1\\ f^{-1}(x)=x^3+1\end{array}$

Hence, the inverse of the given function is $f^{-1}(x)=x^3+1$ .

(b)

To determine

To sketch: the graphs both the functions f and $f^{-1}.$

Explanation of Solution

Given information:

The function $f(x)=\sqrt[3]{x-1}$

Graph:

The graph of the functions $f(x)=\sqrt[3]{x-1},f^{-1}(x)=x^3+1$ are drawn below.

(c)

To determine

To describe: the relation between the graphs of f and $f^{-1}.$

Explanation of Solution

Given information:

The function $f(x)=\sqrt[3]{x-1}$

Consider the graph of part $(b)$

The graph of the $f^{-1}$ is the reflection of graph f about the line $y=x.$

(d)

To determine

To state: The domains and ranges of the functions f and $f^{-1}$ .

Explanation of Solution

Given information:

The function $f(x)=\sqrt[3]{x-1}$

Domain is the set of input values for which the function is defined.

Range is the set of defined output values.

From the graphs of part b,

Consider $f(x)=\sqrt[3]{x-1},f^{-1}(x)=x^3+1$ is

Domain: $-\infty <x<\infty$ , Range: $-\infty <y<\infty$

Hence, the domain of f is range of $f^{-1}$ and range of f is domain of $f^{-1}$