Which statement(s)verify that $f ( x ) = \sqrt [ 7 ] { x + 4 }$ and $g ( x ) = x ^ { 2 } - 4$ are inverse?
A) $f ( g ( x ) ) = \sqrt [ 7 ] { x ^ { 7 } - 4 + 4 } = x ; g ( f ( x ) ) = ( \sqrt [ 7 ] { x } ) ^ { 7 } + 4 - 4 = x$
B) $f ( x ) \cdot g ( x ) = \sqrt [ 7 ] { x + 4 } \cdot \left( x ^ { 7 } - 4 \right) = x$
C) $f ( g ( x ) ) = \sqrt [ 7 ] { x ^ { 7 } } - 4 + 4 = x ; g ( f ( x ) ) = ( \sqrt [ 7 ] { x + 4 } ) ^ { 7 } - 4 = x$
D) $f ( g ( x ) ) = \sqrt [ 7 ] { x ^ { 7 } - 4 + 4 } = x ; g ( f ( x ) ) = ( \sqrt [ 7 ] { x + 4 } ) ^ { 7 } - 4 = x$
E) $\frac { f ( x ) } { g ( x ) } = \frac { \sqrt [ 7 ] { x + 4 } } { x ^ { 7 } - b } = - 1$

Question

Quizplus · Accepted Answer

To show that two functions are inverse functions, we need to show that their composition in both orders is the identity function.

Using choice D, we have:

$f(g(x))=f(x^2-4)=\sqrt[7]{x^2-4+4}=x$

$g(f(x))=g(\sqrt[7]{x+4})=(\sqrt[7]{x+4})^2-4=x$

Therefore, $f(x)$ and $g(x)$ are inverse functions.

Which Statement(s)verify That f(x)=x+47f ( x ) = \sqrt [ 7 ] { x + 4 }f(x)=7x+4​

Which Statement(s)verify That $f ( x ) = \sqrt [ 7 ] { x + 4 }$