Problem 216 - Calculus Volume 1

Question

The function \(C=T(F)=(5 / 9)(F-32)\) converts degrees Fahrenheit to degrees Celsius. a. Find the inverse function \(F=T^{-1}(C)\) b. What is the inverse function used for?

Step-by-Step Solution

Answer

Inverse function found: \( F = \frac{9C + 160}{5} \). It converts Celsius to Fahrenheit.

Step 1: Write the Function to Invert

The function given for converting Fahrenheit to Celsius is \( C = T(F) = \frac{5}{9}(F - 32) \). Our task is to find the inverse function, \( F = T^{-1}(C) \).

Step 2: Rearrange to Solve for F

To find the inverse function, we need to express \( F \) in terms of \( C \). We start by rearranging the original function: **Multiply both sides by 9:** \( 9C = 5(F - 32) \).**Distribute the 5 on the right side:** \( 9C = 5F - 160 \).

Step 3: Solve for F

Now, we solve for \( F \): **Add 160 to both sides:** \( 9C + 160 = 5F \).**Divide every term by 5 to isolate \( F \):** \( F = \frac{9C + 160}{5} \). This equation \( F = \frac{9C + 160}{5} \) is the inverse function \( F = T^{-1}(C) \).

Step 4: Explain the Purpose of the Inverse Function

The inverse function \( F = T^{-1}(C) \) is used to convert temperatures from degrees Celsius back to degrees Fahrenheit. This is useful when starting from a temperature in Celsius and needing the equivalent Fahrenheit measure.

Key Concepts

Inverse Function Application

An inverse function essentially reverses the operation of another function. When you have a function that maps an input to a particular output, the inverse function finds the input that corresponds to a given output from the original function. For example, if we know the Celsius temperature and want to find the equivalent Fahrenheit temperature, we need to use the inverse function. In mathematical terms, if a function is represented as \( y = f(x) \), the inverse will be \( x = f^{-1}(y) \). This inverse function helps revert the output back to the input of the original function. In this specific context, the given function translates Fahrenheit to Celsius, so its inverse translates Celsius back to Fahrenheit. Such inverse relationships are crucial in real-world applications such as conversions between different units of measure, where understanding both the function and its inverse offers a complete view of the conversion process.

Fahrenheit to Celsius Conversion

Converting temperatures between Fahrenheit and Celsius involves a specific mathematical relationship. The function \( C = T(F) = \frac{5}{9}(F - 32) \) expresses how to convert Fahrenheit to Celsius. It's a linear relationship that adjusts the difference between Fahrenheit degrees and the freezing point of water, following a multiplication factor that accounts for the difference in scale size between Fahrenheit and Celsius.Here's how it works:

• Subtract 32 from the Fahrenheit temperature. This accounts for the difference in the freezing points. Fahrenheit measures water freezing at 32 degrees.
• Multiply the result by \( \frac{5}{9} \), adjusting the scale from Fahrenheit to Celsius, since 180 Fahrenheit degrees span the same range as 100 Celsius degrees from freezing to boiling water.

This conversion is a common task in scientific work and international travel, where temperature units may differ based on location or field of study.

Solving Equations

Solving equations involves determining the value of unknown variables. It's about manipulating mathematical statements to isolate and identify the value of these variables. In the context of finding an inverse function, solving equations helps express the output variable from the original function in terms of the input variable to create the inverse. For our exercise:

• Start by multiplying each side of \( C = \frac{5}{9}(F - 32) \) by 9, simplifying to \( 9C = 5(F - 32) \).
• Use distribution to expand: \( 9C = 5F - 160 \).
• Rearrange by adding 160 to both sides: \( 9C + 160 = 5F \).
• Divide by 5 to solve for \( F \), resulting in \( F = \frac{9C + 160}{5} \).

This process shows not just the steps needed but demonstrates the logic and algebraic techniques used to obtain the desired inverse function. Solving equations forms the core of algebra and is vital for deriving relationships in mathematical models.