Chapter 1.6, Problem 110E

To determine

To find : the value of function.

Answer to Problem 110E

The value of function $\left(g^{-1}o{f}^{-1}\right)\left(x\right)$ is $2\sqrt[3]{x+3}$.

Explanation of Solution

Given information :

The functions are $f\left(x\right)=\frac{1}{8}x-3$ and $g\left(x\right)=x^3$.

The function for finding value $\left(g^{-1}o{f}^{-1}\right)\left(x\right)$.

Calculation : to find the value of function, it is needed to first find the inverse functions of $f\left(x\right)\text{ and }g\left(x\right)$.

To find the inverse function of $f\left(x\right)$ , substitute $f\left(x\right)\text{=}y$ in the function,

  $\begin{array}{l}f\left(x\right)=\frac{1}{8}x-3\\ y=\frac{1}{8}x-3\text{ [substitute }f\left(x\right)=y]\\ \frac{1}{8}x=y+3\text{ [add 3 both side]}\\ x=8y+24\text{ [multiply by 8]}\\ f^{-1}\left(x\right)=8x+24\text{ [simplify]}\end{array}$

To find the inverse function of $g\left(x\right)$ , substitute $g\left(x\right)\text{=}y$ in the function,

  $\begin{array}{l}g\left(x\right)=x^3\\ y=x^3\text{ [substitute }g\left(x\right)=y]\\ x=\sqrt[3]{y}\text{ [cube root]}\\ g^{-1}\left(x\right)=\sqrt[3]{x}\text{ [simplify]}\end{array}$

Now to find the value of function $\left(g^{-1}o{f}^{-1}\right)\left(x\right)$ ,

  $\begin{array}{l}\left(g^{-1}o{f}^{-1}\right)\left(x\right)=g^{-1}\left(f^{-1}\left(x\right)\right)\\ \left(g^{-1}o{f}^{-1}\right)\left(x\right)=g^{-1}\left(8x+24\right)\text{ [ }f\left(x\right)=8x+24]\\ \left(g^{-1}o{f}^{-1}\right)\left(x\right)=\sqrt[3]{\left(8x+24\right)}\text{ [substitute }x=\left(8x+24\right)\text{ in }{g}^{-1}\left(x\right)=\sqrt[3]{x}]\\ \left(g^{-1}o{f}^{-1}\right)\left(x\right)=2\sqrt[3]{x+3}\text{ [simplify]}\end{array}$

Thus, The value of function $\left(g^{-1}o{f}^{-1}\right)\left(x\right)$ is $2\sqrt[3]{x+3}$.