surface area of a sphere is S(x) = 4πx2

Question

The surface area of a sphere is S(x) = 4πx², where x is the length of the radius of the sphere. Restrict the domain to create a one-to-one function. Find and describe the inverse function.

Answer 1

S(r) = 4pi(r)^2

to be one-to-one, there can't be 2 values of r for one value of S

so restrict the domain to r> or = 0

or to r< or = 0, but having only a positive radius makes more sense

switch S and r and solve for the new S

r=4piS^2

S^2 = r/4pi

S = sqr(r/4pi)

=.5sqr(r/4pi)

S^-1(r) =[sqr(r/pi)]/2 is the inverse function

(ignore the negative square root as two values of S for one value of r are inconsistent with a on-to-one function)

S^-1(S(r)) = .5sqr(4pir^2/pi) = .5(2)sqr(r^2) = r

S(S^-1(r) = 4pi(.5sqr(r/4pi)^2 = 4pi.25r/4pi = r