The volume of a cylinder whose height is equal to twice its radius is $\mathrm { f } ( \mathrm { x } ) = 2 \pi \mathrm { x } ^ { 3 } \mathrm {~cm} ^ { 3 }$ , where x is the radius of the cylinder in cm . Find the inverse of this function.
A) $f ^ { - 1 } ( x ) = 2 \pi \sqrt [ 3 ] { x }$
B) $f ^ { - 1 } ( x ) = \frac { 1 } { 2 \pi x ^ { 3 } }$
C) $f ^ { - 1 } ( x ) = \sqrt [ 3 ] { \frac { x } { 2 \pi } }$
D) $f ^ { - 1 } ( x ) = \frac { \sqrt [ 3 ] { x } } { 2 \pi }$

Question

The volume of a cylinder whose height is equal to twice its radius is $\mathrm { f } ( \mathrm { x } ) = 2 \pi \mathrm { x } ^ { 3 } \mathrm {~cm} ^ { 3 }$ , where  x  is the radius of the cylinder in  cm . Find the inverse of this function.
A) $f ^ { - 1 } ( x ) = 2 \pi \sqrt [ 3 ] { x }$
B) $f ^ { - 1 } ( x ) = \frac { 1 } { 2 \pi x ^ { 3 } }$
C) $f ^ { - 1 } ( x ) = \sqrt [ 3 ] { \frac { x } { 2 \pi } }$
D) $f ^ { - 1 } ( x ) = \frac { \sqrt [ 3 ] { x } } { 2 \pi }$

Quizplus · Accepted Answer

To find the inverse function, solve for $x$ in terms of $f(x)$. Given $f(x) = 2\pi x^3$, solving for $x$ gives $x = \sqrt[3]{\frac{f(x)}{2\pi}}$, which corresponds to option C when replacing $f(x)$ with $x$ for the inverse function notation.

The Volume of a Cylinder Whose Height Is Equal to Twice