The surface area of S of a sphere of radius r is given by S(r)=4πr2. Explain why, in the given context S(r) has an inverse function.

Question

The surface area of S of a sphere of radius r is given by S(r)=4πr². Explain why, in the given context S(r) has an inverse function.

Identify the correct explanation.

1. S(r) is linear and must have an inverse
2. S(r) is a polynomial and all polynomials are one-to-one for all real numbers
3. S(r) is decreasing on its domain and therefore is one-to-one
4. S(r) is increasing on its domain and therefore is one-to-one
5. S(r) is both increasing and decreasing on its domain.

Find the inverse S(r), expressing r as a function of S

r(S)= ?

Answer 1

The reason that the surface area function S(r)=4πr² has an inverse function stems from the fact that, in the geometric context where the function is applied, r > 0. In interval notation the domain of S(r) is (0,∞).

From the given list, the correct explanation why S(r) has an inverse function is

1. S(r) is increasing on its domain and therefore is one-to-one

To get the inverse function of S = 4πr² we solve for r.

Dividing both sides by 4π gives

r² = S/(4π)

Taking the square root of both sides gives our inverse function

r(S) = √[S/(4π)]