Decide whether or not the functions are inverses of each other.Use the definition of inverses to determine whether f and g are inverses.
-$$f ( x ) = x ^ { 3 } - 6 , \quad g ( x ) = \sqrt [ 3 ] { x + 6 }$$
A) Yes
B) $\mathrm { No }$

Question

Quizplus · Accepted Answer

To determine if $f$ and $g$ are inverses of each other, we need to check if $f(g(x)) = x$ and $g(f(x)) = x$. For $f(x) = x^3 - 6$ and $g(x) = \sqrt[3]{x + 6}$, we find $f(g(x)) = (\sqrt[3]{x + 6})^3 - 6 = x + 6 - 6 = x$ and $g(f(x)) = \sqrt[3]{(x^3 - 6) + 6} = \sqrt[3]{x^3} = x$. Since both compositions return $x$, $f$ and $g$ are indeed inverses of each other.

Decide Whether or Not the Functions Are Inverses of Each f(x)=x3−6,g(x)=x+63f ( x ) = x ^ { 3 } - 6 , \quad g ( x ) = \sqrt [ 3 ] { x + 6 }f(x)=x3−6,g(x)=3x+6​

Decide Whether or Not the Functions Are Inverses of Each $f ( x ) = x ^ { 3 } - 6 , \quad g ( x ) = \sqrt [ 3 ] { x + 6 }$