Chapter 1.6, Problem 115E

To determine

To find : the value of function.

Answer to Problem 115E

The value of function $\left(f^{-1}o{f}^{-1}\right)\left(-6\right)$ is $-168$.

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 $\left(f^{-1}o{f}^{-1}\right)\left(-6\right)$.

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 $\left(f^{-1}o{f}^{-1}\right)\left(-6\right)$ , substitute $x=-6$ ,

  $\begin{array}{l}\left(f^{-1}o{f}^{-1}\right)\left(-6\right)=f^{-1}\left(f^{-1}\left(-6\right)\right)\\ \left(f^{-1}o{f}^{-1}\right)\left(-6\right)=f^{-1}\left(8\left(-6\right)+24\right)\text{ [substitue }x=-6\text{ in }f\left(x\right)=8x+24]\\ \left(f^{-1}o{f}^{-1}\right)\left(-6\right)=f^{-1}\left(-24\right)\text{ [multiply and subtract]}\\ \left(f^{-1}o{f}^{-1}\right)\left(-6\right)=8\left(-24\right)\text{+24 [substitute }x=-24\text{ in }f\left(x\right)=8x+24]\\ \left(f^{-1}o{f}^{-1}\right)\left(-6\right)=-168\text{ [simplify]}\end{array}$

Thus, The value of function $\left(f^{-1}o{f}^{-1}\right)\left(-6\right)$ is $-168$.