Activity 1.7.2.
Recall Dolbear’s function \(F = D(N) = 40 + \frac{1}{4}N\) that converts the number, \(N\text{,}\) of snowy tree cricket chirps per minute to a corresponding Fahrenheit temperature. We have earlier established that the domain of \(D\) is \([40,180]\) and the range of \(D\) is \([50,85]\text{.}\)
(a)
Solve the equation \(F = 40 + \frac{1}{4}N\) for \(N\) in terms of \(F\text{.}\) Call the resulting function \(N = E(F)\text{.}\)
(b)
Explain in words the process or effect of the function \(N = E(F)\text{.}\) What does it take as input? What does it generate as output?
(c)
Use the function \(E\) that you found in (a.) to compute \(j(N) = E(D(N))\text{.}\) Simplify your result as much as possible. Do likewise for \(k(F) = D(E(F))\text{.}\) What do you notice about these two composite functions \(j\) and \(k\text{?}\)
(d)
Consider the equations \(F = 40 + \frac{1}{4}N\) and \(N = 4(F-40)\text{.}\) Do these equations express different relationships between \(F\) and \(N\text{,}\) or do they express the same relationship in two different ways? Explain.
