Q. 92 - Precalculus Enhanced with Graphing Utilities

Question

Temperature Conversion The function $F\left(C\right)=\frac{9}{5}C+32$ converts a temperature from C degrees Celsius to F degrees

Fahrenheit.

(a) Express the temperature in degrees Celsius C as a  function of the temperature in degrees Fahrenheit F.

(b) Verify that $C=C\left(F\right)$ is the inverse of $F=F\left(C\right)$ by showing that $C\left(F\left(C\right)\right)=C$ and $F\left(C\left(F\right)\right)=F$.

(c) What is the temperature in degrees Celsius if it is $70$ degrees Fahrenheit?

Step-by-Step Solution

Answer

Part (a). The expression in degree Celcius as the function of Fahrenheit is $C\left(F\right)=\frac{5}{9}\left(F-32\right)$.

Part (b). It is verified that $F=F\left(C\right)$ is the inverse of $C=C\left(F\right)$.

Part (c). The temperature in degrees Celsius if it is $70$ degrees Fahrenheit is ${21}^{\circ }C$.

Part (a) Step 1. Express the temperature in degrees Celsius as the function of Fahrenheit.

The given function is $F\left(C\right)=\frac{9}{5}C+32$.

Subtract $32$both sides, we get:

$F\left(C\right)-32=\frac{9}{5}C$

Dividing both sides by $\frac{9}{5}$, we get:

$$\begin{gathered} \frac{\left(F-32\right)}{{\displaystyle \frac{9}{5}}}=C\left(F\right) \\ C\left(F\right)=\frac{5}{9}\left(F-32\right) \end{gathered}$$

Part (b) Step 1. Verify the inverse

Substituting $F\left(C\right)$ in the function $C\left(F\left(C\right)\right)=C$, we get:

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

This is true.

Substituting $C\left(F\right)$ in the function $F\left(C\left(F\right)\right)=F$, we get:

$$\begin{gathered} F\left(\frac{5}{9}\left(F-32\right)\right)=F \\ \frac{9}{5}\left(\frac{5}{9}\left(F-32\right)\right)+32=F \\ F-32+32=F \\ F=F \end{gathered}$$

This is true.

Thus, it is verified that $C=C\left(F\right)$ is the inverse of $F=F\left(C\right)$.  

Part (c). Step 1. Find the temperature in Celcius.

Substituting $F=70$ in the function

$C\left(F\right)=\frac{5}{9}\left(F-32\right)$, we get:

$$\begin{gathered} C\left(F\right)=\frac{5}{9}\left(70-32\right) \\ C\left(F\right)=\frac{38\times 5}{9} \\ C\left(F\right)={21}^{\circ }C \end{gathered}$$