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

Question

Quizplus · Accepted Answer

To determine if $(f \circ g)(x) = x$ and $(g \circ f)(x) = x$, we need to compose the functions $f$ and $g$ both ways and simplify.1. $(f \circ g)(x) = f(g(x)) = f(\sqrt[3]{x - 8}) = (\sqrt[3]{x - 8})^3 + 8 = x - 8 + 8 = x$, so $(f \circ g)(x) = x$.2. $(g \circ f)(x) = g(f(x)) = g(x^3 + 8) = \sqrt[3]{(x^3 + 8) - 8} = \sqrt[3]{x^3} = x$, so $(g \circ f)(x) = x$.Both compositions result in the identity function, so the correct answer is that both compositions return $x$.

Determine Whether (F  G)(x) = X and Whether (G f(x)=x3+8,g(x)=x−83f ( x ) = x ^ { 3 } + 8 , g ( x ) = \sqrt [ 3 ] { x - 8 }f(x)=x3+8,g(x)=3x−8​

Determine Whether (F  G)(x) = X and Whether (G $f ( x ) = x ^ { 3 } + 8 , g ( x ) = \sqrt [ 3 ] { x - 8 }$