Preview Activity 2.6.1.
The equation \(y = \frac{5}{9}(x-32)\) relates a temperature given in \(x\) degrees Fahrenheit to the corresponding temperature \(y\) measured in degrees Celsius.
(a)
Solve the equation \(y = \frac{5}{9}(x-32)\) for \(x\) to write \(x\) (Fahrenheit temperature) in terms of \(y\) (Celsius temperature).
(b)
Let \(C(x) = \frac{5}{9}(x-32)\) be the function that takes a Fahrenheit temperature as input and produces the Celsius temperature as output. In addition, let \(F(y)\) be the function that converts a temperature given in \(y\) degrees Celsius to the temperature \(F(y)\) measured in degrees Fahrenheit. Use your work in (a) to write a formula for \(F(y)\text{.}\)
(c)
Next consider the new function defined by \(p(x) = F(C(x))\text{.}\) Use the formulas for \(F\) and \(C\) to determine an expression for \(p(x)\) and simplify this expression as much as possible. What do you observe?
(d)
Now, let \(r(y) = C(F(y))\text{.}\) Use the formulas for \(F\) and \(C\) to determine an expression for \(r(y)\) and simplify this expression as much as possible. What do you observe?
(e)
What is the value of \(C'(x)\text{?}\) of \(F'(y)\text{?}\) How do these values appear to be related?
