Answer
a) $x$
b) $x$
c) $x^9+3x^6+3x^3+2$
Work Step by Step
We are given the functions:
$f(x)=\sqrt[3]{x-1}$
$g(x)=x^3+1$
a) Find $f\circ g$:
$(f\circ g)(x)=f(g(x))=f(x^3+1)=\sqrt[3]{x^3+1-1}=\sqrt[3]{x^3}=x$
b) Find $g\circ f$:
$(g\circ f)(x)=g(f(x))=g(\sqrt[3]{x-1})=(\sqrt[3]{x-1})^3+1=x-1+1=x$
c) Find $g\circ g$:
$(g\circ g)(x)=g(g(x))=g(x^3+1)=(x^3+1)^3+1=x^9+3x^6+3x^3+2$
