Problem 19 - Essential Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The inverse is \(F = \frac{9}{5}C + 32\); its domain is all real numbers.

Step 1: Understand the Formula

The given formula is \(C=\frac{5}{9}(F-32)\), which converts a temperature from Fahrenheit (\(F\)) to Celsius (\(C\)). Our goal is to find the inverse function to convert Celsius back to Fahrenheit.

Step 2: Set Up for Inversion

To find the inverse function, first solve for \(F\) in terms of \(C\). Start by rewriting the equation: \(C = \frac{5}{9}(F - 32)\).

Step 3: Isolate the 'F' term

Multiply both sides by \(\frac{9}{5}\) to cancel out \(\frac{5}{9}\): \[ \frac{9}{5}C = F - 32 \]

Step 4: Solve for ‘F’

Add 32 to both sides to isolate \(F\): \[ F = \frac{9}{5}C + 32 \]This equation represents the inverse function, converting Celsius back to Fahrenheit.

Step 5: Interpret the Inverse Function

The inverse function \(F = \frac{9}{5}C + 32\) shows how to calculate the Fahrenheit temperature given a Celsius temperature. It reverses the effect of the original function.

Step 6: Determine the Domain of the Inverse Function

The domain of the inverse function is the range of the original function. Since \(F\) must be greater than or equal to -459.67, and there are no bounds on \(C\) in the original function, the domain of \(F = \frac{9}{5}C + 32\) is all real numbers, considering no lower bound.

Key Concepts

Celsius to Fahrenheit conversion

Converting temperatures between different units is an essential skill in science and everyday life. The conversion from Celsius to Fahrenheit involves taking the Celsius temperature \(C\) and applying the following formula: \[ F = \frac{9}{5}C + 32 \] This formula is the inverse of the formula to convert Fahrenheit to Celsius, which is originally \(C = \frac{5}{9}(F - 32)\). Understanding these two formulas allows you to move seamlessly between the two temperature scales.When converting Celsius to Fahrenheit:

• First, multiply the Celsius value by \(\frac{9}{5}\).
• Then, add 32 to the result.

This method effectively adjusts the Celsius value to account for the relative difference in the size of the Fahrenheit degree compared to the Celsius degree and aligns the freezing points of water on the two scales. Remembering these steps helps in precise temperature conversions, whether to understand weather forecasts or scientific data.

Function domain

The domain of a function refers to all possible values of the input that will produce a valid output. In the context of temperature conversions, or any function really, understanding the domain is crucial.For the Celsius to Fahrenheit conversion, the function \(C = \frac{5}{9}(F - 32)\) originally restricts the Fahrenheit values to those greater than or equal to -459.67, which is absolute zero, the lowest theoretically possible temperature. This constraint doesn't directly affect Celsius values.When considering the inverse function \(F = \frac{9}{5}C + 32\):

• The domain now refers to the Celsius input values that maintain a valid Fahrenheit output.
• Since there are no specific constraints placed on Celsius in the initial conversion function, the domain here is all real numbers.

Thus, any real number input for Celsius will provide a meaningful corresponding temperature in Fahrenheit, reflecting the wide applicability of this conversion in various contexts.

Solving equations

Solving equations is a key skill in mathematics. It involves manipulating the equation to find the value of one variable in terms of others. This process is essential for deriving functions and their inverses, like in the temperature conversion example. Here's a recap on applying equation-solving skills:To find the inverse for converting Celsius to Fahrenheit, from \(C = \frac{5}{9}(F - 32)\) to \(F = \frac{9}{5}C + 32\), the steps are as follows:

• Identify the variable to isolate, in this case, Fahrenheit \(F\).
• Reverse the operations applied to the original variable (Celsius \(C\)). Here, multiply both sides by \(\frac{9}{5}\) to cancel out the \(\frac{5}{9}\).
• Complete the isolation by solving for \(F\) completely by adding 32.

By systematically reversing the operations and performing similar algebraic manipulations, you can isolate any desired variable, revealing its relationship with others. This skill is versatile and widely applicable in scientific calculations, economic modeling, and beyond.