Given the following function, h( x), find two functions f and g such that $\begin{array} { r }
( f \circ g ) ( x ) = h ( x ) . \
h ( x ) = \sqrt [ 3 ] { x ^ { 2 } - 11 }
\end{array}$
A) $f ( x ) = \sqrt [ 3 ] { x ^ { 2 } } , g ( x ) = - 11$
B) $f ( x ) = \sqrt [ 3 ] { x ^ { 2 } } , g ( x ) = x - 11$
C) $f ( x ) = \sqrt [ 3 ] { x } , g ( x ) = x - 11$
D) $f ( x ) = \sqrt [ 3 ] { x - 11 } , g ( x ) = x ^ { 2 }$
E) $f ( x ) = \sqrt [ 3 ] { x - 11 } , g ( x ) = x + 11$

Question

Given the following function, h( x), find two functions f and g such that $\begin{array} { r } 
( f \circ g ) ( x ) = h ( x ) . \
h ( x ) = \sqrt [ 3 ] { x ^ { 2 } - 11 }
\end{array}$
A) $f ( x ) = \sqrt [ 3 ] { x ^ { 2 } } , g ( x ) = - 11$
B) $f ( x ) = \sqrt [ 3 ] { x ^ { 2 } } , g ( x ) = x - 11$
C) $f ( x ) = \sqrt [ 3 ] { x } , g ( x ) = x - 11$
D) $f ( x ) = \sqrt [ 3 ] { x - 11 } , g ( x ) = x ^ { 2 }$
E) $f ( x ) = \sqrt [ 3 ] { x - 11 } , g ( x ) = x + 11$

Quizplus · Accepted Answer

We need to find two functions f and g such that (f ∘ g)(x) = h(x) = (x^2 - 11)^(1/3). 
Taking f(x) = (x - 11)^(1/3) and g(x) = x^2, we have 
(f ∘ g)(x) = f(g(x)) = f(x^2) = ((x^2) - 11)^(1/3) 
= (x^2 - 11)^(1/3) = h(x). 
Therefore, the correct choice is D.

Given the Following Function, H( X), Find Two Functions F (f∘g)(x)=h(x).h(x)=x2−113\begin{array} { r } ( f \circ g ) ( x ) = h ( x ) . \\h ( x ) = \sqrt [ 3 ] { x ^ { 2 } - 11 }\end{array}(f∘g)(x)=h(x).h(x)=3x2−11​​

Given the Following Function, H( X), Find Two Functions F $\begin{array} { r } ( f \circ g ) ( x ) = h ( x ) . \\h ( x ) = \sqrt [ 3 ] { x ^ { 2 } - 11 }\end{array}$