Chapter 1.5, Problem 21E

Solution

To find: The inverse of $f\left(x\right)=\sqrt[3]{x+5}$ and the domain of $f^{-1}$ .

  $f^{-1}\left(x\right)=x^3-5$ .

  $D\in \left(-\infty ,\infty \right)$ .

Given information:

  $f\left(x\right)=\sqrt[3]{x+5}$

Concept Used:

Domain: It is input value of $x$ where function is well defined.

Calculation:

The domain of $f\left(x\right)=\sqrt[3]{x+5}$ is all real numbers.

  $D\in \left(-\infty ,\infty \right)$

Let $f\left(x\right)=y$

  $\begin{array}{c}y=\sqrt[3]{x+5}\left[\text{switch }x\text{ and }y\right]\\ x=\sqrt[3]{y+5}\left[\text{solve for }y\right]\\ x^3=y+5\\ y=x^3-5\left[y=f^{-1}\left(x\right)\right]\\ f^{-1}\left(x\right)=x^3-5\end{array}$

The inverse of function is $f^{-1}\left(x\right)=x^3-5$ and the domain is all real numbers. $D\in \left(-\infty ,\infty \right)$ .