Chapter 1.5, Problem 73E

To determine

To identify two functions $f$ and $g$ such that $\left(f\circ g\right)\left(x\right)=h\left(x\right)$ for the given function $h\left(x\right)$ .

Answer to Problem 73E

  $f\left(x\right)=\sqrt[3]{x}$ , $g\left(x\right)=x^2-4$

Explanation of Solution

Given:

The function $h\left(x\right)=\sqrt[3]{x^2-4}$ .

Calculation:

For the given function $h\left(x\right)$ , the inner function is $g\left(x\right)=x^2-4$ . So, the outer function will be $f\left(x\right)=\sqrt[3]{x}$ .

Now, find $\left(f\circ g\right)\left(x\right)$ to check if the two functions give $h\left(x\right)$ or not.

  $\begin{array}{c}\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)\\ =f\left(x^2-4\right)\\ =\sqrt[3]{x^2-4}\\ =h\left(x\right)\end{array}$

Thus, the two functions are $f\left(x\right)=\sqrt[3]{x}$ and $g\left(x\right)=x^2-4$ .