Chapter 1.9, Problem 46E

(a)

To determine

To find: inverse of the function of f .

Answer to Problem 46E

The inverse of the given function is $f^{-1}(x)=\sqrt[3]{x-8}$ .

Explanation of Solution

Given information:

The function $f(x)=x^3+8$

Calculation:

Let the function $y=f(x)=x^3+8$

  $y=x^3+8$

  $\begin{array}{l}y-8=x^3\\ x=\sqrt[3]{y-8}\\ f^{-1}(y)=\sqrt[3]{y-8}\\ f^{-1}(x)=\sqrt[3]{x-8}\end{array}$

Hence, the inverse of the given function is $f^{-1}(x)=\sqrt[3]{x-8}$ .

(b)

To determine

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

Explanation of Solution

Given information:

The function $f(x)=x^3+8$

Graph:

The graph of the functions $f(x)=x^3+8$ and $f^{-1}(x)=\sqrt[3]{x-8}$ is drawn below.

(c)

To determine

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

Explanation of Solution

Given information:

The function $f(x)=x^3+8$

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 $f(x)=x^3+8$

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 $f(x)=x^3+8$ is

Domain: all real numbers

Range: all the real numbers.

Consider $f^{-1}(x)=\sqrt[3]{x-8}$ is

Domain: all real numbers

Range: all the real numbers.

Hence, the domains and ranges of the functions f and $f^{-1}$ is all real numbers.