Chapter 2 - 2.7 - Inverse Functions - 2.7 Exercises - Page 229: 84

$g^{-1}~o~f^{-1}=2\sqrt[3] {x+3}$

$f(x)=\frac{1}{8}x-3$ $y=\frac{1}{8}x-3~~$ (Interchange $x$ and $y$): $x=\frac{1}{8}y-3$ $x+3=\frac{1}{8}y$ $y=8x+24$ $f^{-1}(x)=8x+24$ $g(x)=x^3$ $y=x^3~~$ (Interchange $x$ and $y$): $x=y^3$ $y=\sqrt[3] x$ $g^{-1}(x)=\sqrt[3] x$ $g^{-1}~o~f^{-1}=g^{-1}(f^{-1})=g^{-1}(8x+24)=\sqrt[3] {8x+24}=\sqrt[3] {8(x+3)}=2\sqrt[3] {x+3}$