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.

Why I'm Taking BCUSP 123

At age 46, I find myself back in the classroom as a student for the first time in 24 years. My kids are growing up and I'm trying to redefine myself for the next phase of my life. Before I had kids I was a Learning Disabilities specialist, and I've decided to return to the workforce in teaching once more. I'm enrolled in the UWB K-8 Teacher Certification program, with the additional goal of acquiring a Middle Level endorsement in Math. Although I always enjoyed math and took classes through multi-dimensional calculus in college, I'm more than a little rusty. My goals in taking this class are to review some content, experience being a math student again, and acquaint myself with some newer math technology.

I actually really like math. I was a product of the 1970s push for individualized learning, so I self-taught math for several years between about 5th and 8th grade. We all just worked out way through sets of cards, and took tests at the end of each section. In high school and college I had more conventional classes. I suppose I like the problem-solving aspect of mathematics. I do remember that I did not particularly like geometry proofs.

Quadratic equations...Well, they are second degree equations. They typically take the form a*x(squared) + bx + c = 0. (Sorry, I don't know how to do superscript!) They can be solved using several different techniques, including factoring, completing the square, and using the quadratic formula. When they are graphed they look like parabolas.