The surface area of a sphere is $f ( x ) = 4 \pi x ^ { 2 }$ square inches, where x is the radius of the sphere in inches. Find the inverse of this function.
A) $f ^ { - 1 } ( x ) = \frac { \sqrt { x } } { 2 \pi }$
B) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 2 } \sqrt { \frac { \mathrm { x } } { \pi } }$
C) $f ^ { - 1 } ( x ) = 2 \sqrt { \frac { \pi } { x } }$
D) $f ^ { - 1 } ( x ) = 2 \sqrt { \pi x }$

Question

The surface area of a sphere is $f ( x ) = 4 \pi x ^ { 2 }$ square inches, where  x  is the radius of the sphere in inches. Find the inverse of this function.
A) $f ^ { - 1 } ( x ) = \frac { \sqrt { x } } { 2 \pi }$
B) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 2 } \sqrt { \frac { \mathrm { x } } { \pi } }$
C) $f ^ { - 1 } ( x ) = 2 \sqrt { \frac { \pi } { x } }$
D) $f ^ { - 1 } ( x ) = 2 \sqrt { \pi x }$

Quizplus · Accepted Answer

To find the inverse function, you swap $x$ and $y$ and solve for $y$. Starting with $y = 4\pi x^2$, swap to get $x = 4\pi y^2$, and solve for $y$: $y = \frac{1}{2}\sqrt{\frac{x}{\pi}}$.

The Surface Area of a Sphere Is f(x)=4πx2f ( x ) = 4 \pi x ^ { 2 }f(x)=4πx2

The Surface Area of a Sphere Is $f ( x ) = 4 \pi x ^ { 2 }$