Question

The formula \(C = \frac{5}{9}\left( {F - 32} \right)\), where \(F \ge - 459.67,\) expenses the Celsius temperature \(C\) as a function of the Fahrenheit temperature \(F\). Find a formula for the inverse function and interpret it. What is the domain of the inverse function?

Short answer

The formula \(F = \frac{5}{9}C + 32\) provides us a formula for the inverse function, which gives us Fahrenheit temperature \(F\) as a function of the Celsius temperature \(C\). The domain of the inverse function is \(\left( { - 273.15,\infty } \right)\).

Step-by-step solution

Condition for the inverse function

Consider \(f\) as the one-to-one functionwith the domain \(A\) and range \(B\). Then its inverse function \({f^{ - 1}}\)has domain \(B\) and range \(A\) and is defined as

\({f^{ - 1}}\left( y \right) = x \Leftrightarrow f\left( x \right) = y\) for any \(y\) in \(B\).

Determine the formula for the inverse function and interpret it

Solve the formula for \(F\) as shown below:

\(\begin{aligned}C &= \frac{5}{9}\left( {F - 32} \right)\\\frac{9}{5}C &= F - 32\\F &= \frac{9}{5}C + 32\end{aligned}\)

This provides us a formula for the inverse function, which gives us Fahrenheit temperature \(F\) as a function of the Celsius temperature \(C\).

Determine the domain of the inverse function

The domain is:

\(\begin{aligned}F \ge - 459.67\\\frac{9}{5}C + 32 \ge - 459.67\\\frac{9}{5}C \ge - 491.67\\C \ge - 273.15\end{aligned}\)

Thus, the domain of the inverse function is \(\left( { - 273.15,\infty } \right)\).