Preview Activity 1.7.1.
Recall that \(F = g(C) = \frac{9}{5}C + 32\) is the function that takes Celsius temperature inputs and produces the corresponding Fahrenheit temperature outputs.
(a)
Show that it is possible to solve the equation \(F = \frac{9}{5}C + 32\) for \(C\) in terms of \(F\) and that doing so results in the equation \(C = \frac{5}{9}(F-32)\text{.}\)
(b)
Note that the equation \(C = \frac{5}{9}(F-32)\) expresses \(C\) as a function of \(F\text{.}\) Call this function \(h\) so that \(C = h(F) = \frac{5}{9}(F-32)\text{.}\)
Find the simplest expression that you can for the composite function \(j(C) = h(g(C))\text{.}\)
(c)
Find the simplest expression that you can for the composite function \(k(F) = g(h(F))\text{.}\)
(d)
Why are the functions \(j\) and \(k\) so simple? Explain by discussing how the functions \(g\) and \(h\) process inputs to generate outputs and what happens when we first execute one followed by the other.
