Decide whether or not the functions are inverses of each other.
-$f ( x ) = \sqrt [ 3 ] { 3 x - 7 } , g ( x ) = \frac { x ^ { 3 } + 7 } { 3 }$
A) No
B) Yes

Question

Quizplus · Accepted Answer

To determine if two functions are inverses of each other, we can compose them (i.e., substitute one function into the other) and see if the result is the identity function, $x$. Let's compose $f(g(x))$ and $g(f(x))$.First, $f(g(x)) = f\left(\frac{x^3 + 7}{3}\right) = \sqrt[3]{3\left(\frac{x^3 + 7}{3}\right) - 7} = \sqrt[3]{x^3 + 7 - 7} = \sqrt[3]{x^3} = x$.Next, $g(f(x)) = g\left(\sqrt[3]{3x - 7}\right) = \frac{\left(\sqrt[3]{3x - 7}\right)^3 + 7}{3} = \frac{3x - 7 + 7}{3} = \frac{3x}{3} = x$.Since both compositions result in the identity function, $f(x)$ and $g(x)$ are indeed inverses of each other.

Decide Whether or Not the Functions Are Inverses of Each f(x)=3x−73,g(x)=x3+73f ( x ) = \sqrt [ 3 ] { 3 x - 7 } , g ( x ) = \frac { x ^ { 3 } + 7 } { 3 }f(x)=33x−7​,g(x)=3x3+7​

Decide Whether or Not the Functions Are Inverses of Each $f ( x ) = \sqrt [ 3 ] { 3 x - 7 } , g ( x ) = \frac { x ^ { 3 } + 7 } { 3 }$