Problem 50 - Calculus of a Single Variable

Question

The formula \(C=\frac{5}{9}(F-32),\) where \(F \geq-459.6\) , represents Celsius temperature \(C\) as a function of Fahrenheit temperature \(F .\) (a) Find the inverse function of \(C .\) (b) What does the inverse function represent? (c) What is the domain of the inverse function? Validate or explain your answer using the context of the problem. (d) The temperature is \(22^{\circ} \mathrm{C}\) . What is the corresponding temperature in degrees Fahrenheit?

Step-by-Step Solution

Answer

The inverse function is \(C=\frac{9F+160}{5}\), which represents the conversion of Celsius to Fahrenheit. The domain of this inverse function is realistically \([-273.15, \infty)\) considering absolute zero. A temperature of \(22^{\circ} \mathrm{C}\) equals approximately \(71.6^{\circ} \mathrm{F}\).

Step 1: Finding the Inverse Function of C

We need to find the inverse of the function \(C=\frac{5}{9}(F-32)\). We start by swapping \(C\) and \(F\) to get \(F=\frac{5}{9}(C-32)\). Then, we solve for \(C\), which gives us the inverse function.

Step 2: Solving for C

Multiply by 9 to get rid of the fraction: \(9F=5(C-32)\). Next, distribute 5 to obtain \(9F=5C-160\). Finally, solve for \(C\) by adding 160 and then dividing by 5: \(C=\frac{9F+160}{5}\). This is the inverse function.

Step 3: Interpret the Inverse Function

The inverse function \(C=\frac{9F+160}{5}\) represents the conversion from Celsius to Fahrenheit. This is because this function gives us a Fahrenheit temperature when a Celsius temperature is inputted.

Step 4: Find the Domain of the Inverse Function

Since any Celsius temperature can be converted to Fahrenheit, the domain of the inverse function is all real numbers. However, it is important to consider that absolute zero (the lowest possible temperature) is -273.15 degrees Celsius. Therefore, it would make sense for the domain to be \([-273.15, \infty)\).

Step 5: Convert Temperature from Celsius to Fahrenheit Using the Inverse Function

We have the current temperature as \(22^{\circ} \mathrm{C}\). Substituting this into our inverse function returns the equivalent Fahrenheit temperature: \(F=\frac{9×22+160}{5}=71.6\). So, \(22^{\circ} \mathrm{C}\) is approximately \(71.6^{\circ} \mathrm{F}\).

Key Concepts

Celsius to Fahrenheit Conversion

Understanding the Celsius to Fahrenheit conversion is crucial for various scientific, culinary, and everyday purposes. To perform this temperature conversion, a specific formula is used, which reflects the mathematical relationship between the two temperature scales.

The formula to convert Celsius to Fahrenheit is: \( F = \frac{9}{5}C + 32 \) where \( F \) represents the temperature in degrees Fahrenheit and \( C \) represents the temperature in degrees Celsius.

By using this formula, you can convert any given Celsius temperature to Fahrenheit. For instance, if you need to convert 22 degrees Celsius to Fahrenheit, you substitute 22 for \( C \) in the formula to get: \( F = \frac{9}{5} \times 22 + 32 \) which yields \( F = 71.6 \). Thus, 22 degrees Celsius is equivalent to 71.6 degrees Fahrenheit.

Temperature Conversion Inverse

The inverse of a temperature conversion function allows us to convert from Fahrenheit back to Celsius. This is particularly useful when we have a measurement in Fahrenheit and need to know the corresponding temperature in Celsius.

The inverse function of the original Celsius to Fahrenheit conversion is derived by following algebraic steps to solve for \( C \) given a Fahrenheit temperature \( F \). The inverse function is: \( C = \frac{5}{9}(F - 32) \)

To apply this inverse function, simply plug in the Fahrenheit temperature into the \( F \) variable. For example, to find the Celsius equivalent of 71.6 degrees Fahrenheit, the calculation would be: \( C = \frac{5}{9}(71.6 - 32) \) which gives us the original 22 degrees Celsius. This function provides a systematic method for reversing the Celsius to Fahrenheit conversion.

Domain of Inverse Function

The domain of an inverse function is a set of all possible input values that the function can accept. For temperature conversion, this relates to the range of temperatures that can be converted from Fahrenheit to Celsius.

In the context of temperature, it's important to note that the lowest limit of the Celsius scale is absolute zero, which is -273.15°C. This corresponds to -459.67°F, the lowest limit on the Fahrenheit scale.

Therefore, for the inverse function \( C = \frac{5}{9}(F - 32) \) converting Fahrenheit to Celsius, the domain is theoretically all Fahrenheit temperatures above this absolute zero point. In other words, the domain is \( F \geq -459.67 \) or, in interval notation, \( [-459.67, \infty) \).

This domain consideration ensures that temperature conversions using the inverse function remain physically meaningful and are restricted to temperatures that can actually exist in reality.