Decide whether the composite functions, f ° g and g ° f, are equal to x.
-$$f ( x ) = \sqrt [ 5 ] { x - 8 } , g ( x ) = x ^ { 5 } + 8$$
A) Yes, yes
B) Yes, no
C) No, no
D) No, yes

Question

Quizplus · Accepted Answer

For $f(g(x))$, substituting $g(x) = x^5 + 8$ into $f(x)$, we get $f(g(x)) = \sqrt[5]{(x^5 + 8) - 8} = \sqrt[5]{x^5} = x$. For $g(f(x))$, substituting $f(x) = \sqrt[5]{x - 8}$ into $g(x)$, we get $g(f(x)) = (\sqrt[5]{x - 8})^5 + 8 = x - 8 + 8 = x$. Therefore, both composite functions are equal to $x$.

Decide Whether the Composite Functions, F ° G and G f(x)=x−85,g(x)=x5+8f ( x ) = \sqrt [ 5 ] { x - 8 } , g ( x ) = x ^ { 5 } + 8f(x)=5x−8​,g(x)=x5+8

Decide Whether the Composite Functions, F ° G and G $f ( x ) = \sqrt [ 5 ] { x - 8 } , g ( x ) = x ^ { 5 } + 8$