Thinking about Inverse Functions

I've been working on section 2.4 homework and I'm having trouble reconciling what I thought I understood about inverse functions with the materials on pp. 80-81 in the text. I'll admit I've never really considered real life examples of inverse functions before. In the case of population varying as a function of time (P=f(t)), the text seems to be saying that whereas the original function allows us to find population if we know time, the inverse allows us to find time if we know population. Isn't that just a matter of solving for "X" so to speak?

In the cricket example on p.81, the original function is T = 1/4(R) + 40. Time equals 1/4*rate of chirps + 40. The example goes on to show that the inverse is R = 4(T-40). Maybe I am missing something obvious, but it sure seems to me that all they have done is to solve for R.

I would calculate the inverse of the original cricket function like this:
T = 1/4(R) +40 (original equation)
R = 1/4(T) +40 (exchanging input and output values)
1/4 (T) = R-40
T = 4(R-40) or T = 4R - 160

I don't know how to make sense of this inverse with regard to crickets, but I'm pretty sure this new function is the mathematical inverse of the original function.

I look forward to your clarification on this matter.