Determine whether $$( f \circ g ) ( x ) = x \text { and whether } ( g \circ f ) ( x ) = x \text {. }$$
-$f ( x ) = x ^ { 3 } + 8 , g ( x ) = \sqrt [ 3 ] { x - 8 }$
A) No, yes
B) Yes, yes
C) No, no
D) Yes, no

Question

Quizplus · Accepted Answer

The composition $ ( f \circ g ) ( x ) = f(g(x)) = f(\sqrt[3]{x-8}) = (\sqrt[3]{x-8})^3 + 8 = x - 8 + 8 = x $.  The composition $ ( g \circ f ) ( x ) = g(f(x)) = g(x^3 + 8) = \sqrt[3]{(x^3 + 8) - 8} = \sqrt[3]{x^3} = x $.  Both compositions simplify to $ x $, so both are true.

Determine Whether (f∘g)(x)=x and whether (g∘f)(x)=x. ( f \circ g ) ( x ) = x \text { and whether } ( g \circ f ) ( x ) = x \text {. }(f∘g)(x)=x and whether (g∘f)(x)=x.

Determine Whether $( f \circ g ) ( x ) = x \text { and whether } ( g \circ f ) ( x ) = x \text {. }$