Question

a rule for an example of a nonlinear function that fits the following description. when d is 4, r is 9, and r is a function of d

Answer 1

Some possibilities:
r = (9√d)/2, so f(d) = (9√d)/2
r = 9d^2/16, so f(d) = 9d^2/16
r = 9(2^d)/16, so f(d) = 9(2^d)/16

These are examples of non-linear functions of d which fit the description. I think that is what you are asking, but comment if I'm wrong?