Chapter 1.4, Problem 20E

Solution

To find: The composite function of two functions and find its domain.

The required functions are $f\left(g\left(x\right)\right)=1-x^3$ , domain $ℝ\text{or}\left(-\infty ,\infty \right)$ and $g\left(f\left(x\right)\right)=\sqrt[3]{1-x^6}$ , domain $ℝ\text{or}\left(-\infty ,\infty \right)$ .

Given information: The two functions are $f\left(x\right)=x^3$ and $g\left(x\right)=\sqrt[3]{1-x^3}$ . The required composite functions are $f\left(g\left(x\right)\right)$ and $g\left(f\left(x\right)\right)$ .

Method/Formula used:

The composite functions of two functions $f\left(x\right)$ and $g\left(x\right)$ are denoted by $\left(fog\right)\left(x\right)$ and $\left(gof\right)\left(x\right)$ , and are given by $\left(fog\right)\left(x\right)=f\left(g\left(x\right)\right)$ , and $\left(gof\right)\left(x\right)=g\left(f\left(x\right)\right)$ .

  

Calculations:

The given functions are; $f\left(x\right)=x^3$ and $g\left(x\right)=\sqrt[3]{1-x^3}$ . Therefore,

  $\begin{array}{c}f\left(g\left(x\right)\right)=f\left(g\left(x\right)\right)\leftarrow \text{substitute}{\left(1-x^3\right)}^{\frac{1}{3}}\text{for}g\left(x\right)\\ =f\left({\left(1-x^3\right)}^{\frac{1}{3}}\right)\leftarrow \text{use}\text{definition}\text{of}f\left(x\right)=x^3\\ ={\left({\left(1-x^3\right)}^{\frac{1}{3}}\right)}^3\leftarrow \text{simplify}\\ =1-x^3\end{array}$

Since, the function $f\left(g\left(x\right)\right)=1-x^3$ is defined for all (set of real numbers). Therefore, $ℝ\text{or}\left(-\infty ,\infty \right)$ is the domain of the function $f\left(g\left(x\right)\right)$ .

Further, $f\left(x\right)=x^3$ and $g\left(x\right)=\sqrt[3]{1-x^3}$ .

  $\begin{array}{c}g\left(f\left(x\right)\right)=g\left(f\left(x\right)\right)\leftarrow \text{substitute}{x}^3\text{ for}f\left(x\right)\\ =g\left(x^3\right)\leftarrow \text{use}\text{definition}\text{of}g\left(x\right)={\left(1-x^2\right)}^{\frac{1}{3}}\\ ={\left(1-{\left(x^3\right)}^2\right)}^{\frac{1}{3}}\leftarrow \text{simplify}\\ ={\left(1-x^6\right)}^{\frac{1}{3}}\\ =\sqrt[3]{\left(1-x^6\right)}\end{array}$

Since, the function $g\left(f\left(x\right)\right)=\sqrt[3]{1-x^6}$ is defined for all $x\in ℝ$ (set of real numbers), therefore, $\left(-\infty ,\infty \right]$ is the domain of the function $g\left(f\left(x\right)\right)$ .