Determine whether (f(g(x)) $$( f ( g ( x ) ) = x \text { and whether } ( g ( f ( x ) ) = x$$
-$f ( x ) = x ^ { 3 } + 9 , g ( x ) = \sqrt [ 3 ] { x - 9 }$
A) Yes, no
B) No, no
C) Yes, yes
D) No, yes

Question

Quizplus · Accepted Answer

First, we find $f(g(x)) = f(\sqrt[3]{x-9}) = (\sqrt[3]{x-9})^3 + 9 = x - 9 + 9 = x$, showing that $f(g(x)) = x$. Then, we find $g(f(x)) = g(x^3 + 9) = \sqrt[3]{(x^3 + 9) - 9} = \sqrt[3]{x^3} = x$, showing that $g(f(x)) = x$. Therefore, both compositions return the original input $x$, confirming the answer is yes for both.

Determine Whether (F(g(x)) (f(g(x))=x and whether (g(f(x))=x( f ( g ( x ) ) = x \text { and whether } ( g ( f ( x ) ) = x(f(g(x))=x and whether (g(f(x))=x

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