Question

The formula \(\quad C=\frac{5}{9}(F-32),\) where \(F \geq-459.6,\) represents the 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) Determine the domain of the inverse function. (d) The temperature is \(22^{\circ} \mathrm{C}\). What is the corresponding temperature in degrees Fahrenheit?

Short answer

The inverse function is \(F=\frac{9}{5}C+32\), which represents the Fahrenheit temperature as a function of Celsius temperature. The domain of this inverse function is \([-273.15, \infty)\). Therefore, the temperature \(22^{\circ} \mathrm{C}\) translates to \(71.6^{\circ} \mathrm{F}\).

Step-by-step solution

Find the inverse function

To find the inverse of the function, first replace \(C\) with \(y\), so the initial function looks like \(y=\frac{5}{9}(F-32)\). To get the inverse, swap \(y\) and \(F\). This gives \(F=\frac{5}{9}(y-32)\). Solve this equation for \(y\) to get the inverse function. Multiply every term by \(\frac{9}{5}\) and then add 32 to both sides: \(y=\frac{9}{5}F + 32\). The inverse function, defined as \(F(y)\), is, therefore, \(F(y)=\frac{9}{5}y + 32\).

Describe the inverse function

The inverse function represents the conversion of temperatures from Celsius to Fahrenheit. Given a temperature in Celsius, the inverse function, \(F(y)\), allows us to find the equivalent Fahrenheit temperature.

Determine the Domain of the inverse function

The domain of the inverse function is the set of all valid input values. The original function is defined for all \(F \geq -459.6\) as the Fahrenheit scale goes down only till absolute zero i.e. -459.6°F, equivalent to 0K or -273.15°C in other temperature scales. Hence the domain for the inverse function, which is in Celsius, will be all \(y \geq -273.15\). Therefore, the domain of \(F(y)\) is \([-273.15, \infty)\).

Convert temperature from Celsius to Fahrenheit

The temperature given is \(22^{\circ} \mathrm{C}\), then use the inverse function \(F(y)=\frac{9}{5}y+32\), replacing \(y\) by \(22\): The output (\(F\)) is given by \(F=\frac{9}{5}(22) + 32 = 71.6\).

Key concepts

Temperature Conversion

Understanding temperature conversion is essential in science, travel, and even cooking. The formula used to convert Celsius to Fahrenheit is a practical example of a function in mathematics. To convert temperatures, we use the relationship between the two units, which involves a scaling factor and an addition or subtraction. The formula for converting Celsius (\(C\text{°C}\)) to Fahrenheit (\(F\text{°F}\)) is \( F = \frac{9}{5}C + 32 \).

For instance, if you need to convert 25°C to Fahrenheit, you plug 25 into the place of 'C' in the formula to get \( F = \frac{9}{5}(25) + 32 = 77\text{°F} \). This linear function shows a direct proportionality between the two scales with a slope of \( \frac{9}{5} \) and a y-intercept at 32°. A deep understanding of this function reveals the relationship between both temperature scales and enables easy conversions in practical scenarios.

Celsius to Fahrenheit

When we talk about converting Celsius to Fahrenheit, we are referring to finding the inverse of the Celsius temperature function. The formula used to represent Celsius as a function of Fahrenheit is \( C = \frac{5}{9}(F - 32) \). To find its inverse, we interchange the roles of 'C' and 'F', and solve for the new 'C', now denoted as the variable 'y'.

We arrive at \( y = \frac{9}{5}F + 32 \), where 'y' represents the temperature in Fahrenheit when given a temperature in Celsius. This process illustrates the concept of an inverse function in mathematics, where the output of the original function becomes the input of the inverse, and vice versa, enabling the reverse calculation from one scale to the other.

Function Domain

The domain of a function is the complete set of possible values of the independent variable. In the context of temperature conversion, the domain of the original function, \( F \), relates to Fahrenheit temperatures where it's physically possible, which is \( F \) greater than or equal to -459.6°F (the lowest theoretical temperature, absolute zero).

For the inverse function, which converts Fahrenheit to Celsius, its domain is based on the lowest Celsius temperature that corresponds to the Fahrenheit absolute zero, which is -273.15°C. Thus, the domain for the inverse function is from -273.15°C to positive infinity, represented mathematically as \( \text{domain} = [-273.15, \infty) \). Understanding function domains is crucial as it limits the scope of our function to only those values where the function 'makes sense' and yields meaningful outputs.