Problem 45 - College Algebra

Question

To convert from \(x\) degrees Celsius to \(y\) degrees Fahrenheit, we use the formula \(f(x)=\frac{9}{5} x+32\) Find the inverse function, if it exists, and explain its meaning.

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(y) = \frac{5}{9}(y - 32)\). It converts Fahrenheit to Celsius.

Step 1: Understand the Problem

To find the inverse function of \(f(x) = \frac{9}{5}x + 32\), we need to swap the roles of \(x\) and \(f(x)\) to solve for \(x\) in terms of \(f(x)\). The inverse function will help us convert a temperature from Fahrenheit to Celsius.

Step 2: Introduce Notation for Inverse Function

Let \(y=f(x)\), which implies \(y=\frac{9}{5}x + 32\). We want to express \(x\) in terms of \(y\) to find \(f^{-1}(y)\).

Step 3: Solve for x

Rearrange the equation \(y=\frac{9}{5}x + 32\) to isolate \(x\):1. Subtract 32 from both sides: \(y - 32 = \frac{9}{5}x\)2. Multiply both sides by \(\frac{5}{9}\) to solve for \(x\):\[x = \frac{5}{9}(y - 32)\]

Step 4: Write the Inverse Function

The inverse function \(f^{-1}(y)\) is given by \[f^{-1}(y) = \frac{5}{9}(y - 32)\].This function converts temperatures from Fahrenheit to Celsius.

Step 5: Verify the Inverse Function

To verify that \(f^{-1}(y)\) is indeed the inverse, substitute \(f(x)\) into our inverse function:\[f^{-1}(f(x)) = \frac{5}{9}(\frac{9}{5}x + 32 - 32) = x\] and \[f(f^{-1}(y)) = \frac{9}{5}(\frac{5}{9}(y - 32)) + 32 = y\], confirming that the inverse function correctly converts back to the original value.

Step 6: Interpretation of Inverse Function

The inverse function \(f^{-1}(y) = \frac{5}{9}(y - 32)\) is meaningful because it allows for conversion of temperature from Fahrenheit back to Celsius. This is useful to reverse the direction of the temperature conversion.

Key Concepts

Conversion between Celsius and Fahrenheit

Converting between Celsius and Fahrenheit is a typical task when dealing with temperature measurements. Fahrenheit is mainly used in the United States, whereas Celsius is the preferred scale in most other countries. Understanding how to convert between these two is useful because it allows communication in temperature across different systems.

To express temperature in Fahrenheit from Celsius, a specific linear relationship is used. The formula is:

• To convert Celsius to Fahrenheit: Multiply the Celsius temperature by \(\frac{9}{5}\) and then add 32.
• To revert Fahrenheit to Celsius: The process involves working in reverse or using the inverse function.

This conversion process is quite straightforward once you get the hang of the formula. With this understanding, you can easily switch between these two temperature scales for any scenario.

Temperature Conversion Formula

The temperature conversion formula between Celsius and Fahrenheit is an essential part of this exercise. The function given is \(f(x) = \frac{9}{5}x + 32\), where \(x\) represents the temperature in Celsius, and \(f(x)\) gives the temperature in Fahrenheit.

This formula is derived based on the fixed points of water freezing and boiling in both scales. Water freezes at 0 degrees Celsius, equivalent to 32 degrees Fahrenheit. Similarly, water boils at 100 degrees Celsius, which is 212 degrees Fahrenheit. This relationship is linear and non-proportional due to the 32-degree shift.

Remember, this formula is crucial because it helps measure temperature in various regions. Whether you are traveling or conducting scientific experiments, having this temperature conversion at your fingertips can be incredibly helpful.

Algebraic Inverse Functions

Understanding algebraic inverse functions can demystify many concepts in mathematics, including temperature conversion. An inverse function essentially reverses the effect of another function. If you have a function that takes your input to an output, its inverse will take that output back to the input.

For our problem, the function \(f(x) = \frac{9}{5}x + 32\) converts Celsius to Fahrenheit, and we need its inverse \(f^{-1}(y)\) to convert Fahrenheit back to Celsius.

The process of finding this inverse involves:

• Swapping the input, \(x\), and output, \(y\), and solving for \(x\).
• Rearranging the equation to isolate \(x\), leading to \(x = \frac{5}{9}(y - 32)\), which gives the inverse \(f^{-1}(y)\).

This inverse is fundamental when you need to know the original amount from the transformed state, like changing a known Fahrenheit temperature to Celsius. Thus, inverse functions are powerful tools in algebra, reversing operations and making conversions seamless.