Chapter 1.9, Problem 48E

(a)

To determine

The inverse function of $f$ .

Answer to Problem 48E

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

Explanation of Solution

Given information:

The given function is $f\left(x\right)=x^3+1$ .

Calculation:

The given function is $f\left(x\right)=x^3+1$ .

Replace $f\left(x\right)$ by $y$ and interchange the $x$ and $y$ .

  $\begin{array}{l}y=x^3+1\\ x=y^3+1\end{array}$

Replace $y$ by $f^{-1}\left(x\right)$ .

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

Therefore, the inverse function of $f$ is $f^{-1}\left(x\right)=\sqrt[3]{x-1}$ .

(b)

To determine

The graph of both function $f$ and $f^{-1}$ on same set of coordinate axes.

Answer to Problem 48E

The graph of both function $f$ and $f^{-1}$ is shown in figure (1).

Explanation of Solution

Given information:

The given function is $f\left(x\right)=x^3+1$ .

Calculation:

The function are $f\left(x\right)=x^3+1$ and $f^{-1}\left(x\right)=\sqrt[3]{x-1}$ .

Let graph of both function $f$ and $f^{-1}$ is shown in figure (1).

  

Figure (1)

Therefore, the graph of both function $f$ and $f^{-1}$ is shown in figure (1).

(c)

To determine

The relationship between graphs of $f$ and $f^{-1}$ .

Answer to Problem 48E

The graph of $f^{-1}$ is reflection of $f$ .

Explanation of Solution

Given information:

The given function is $f\left(x\right)=x^3+1$ .

Calculation:

The graph of $f^{-1}$ is reflection of $f$ .

Therefore, the graph of $f^{-1}$ is reflection of $f$ .

(d)

To determine

The domain and range of $f$ and $f^{-1}$ .

Answer to Problem 48E

The domain and range of $f$ is all real numbers and domain of $f^{-1}$ is $\left(1,\infty \right)$ and range of the function is $\left(0,\infty \right)$ .

Explanation of Solution

Given information:

The given function is $f\left(x\right)=x^3+1$ .

Calculation:

The function $f\left(x\right)=x^3+1$ is defined for real value of $x$ and attains all real values.

Thedomain and range of $f$ is all real numbers.

The function $f^{-1}\left(x\right)=\sqrt[3]{x-1}$ is defined for all values $x-1>0$ that are for $x>1$ .

The domain of $f^{-1}$ is $\left(1,\infty \right)$ and range of the function is $\left(0,\infty \right)$ .

Therefore, the domain and range of $f$ is all real numbers and domain of $f^{-1}$ is $\left(1,\infty \right)$ and range of the function is $\left(0,\infty \right)$ .