Problem 45 - College Algebra with Corequisite Support

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 \( f^{-1}(y) = \frac{5}{9}(y - 32) \), converting Fahrenheit to Celsius.

Step 1: Understand the Given Function

We start with the given function for converting Celsius to Fahrenheit: \( f(x) = \frac{9}{5}x + 32 \). Our task is to find the inverse function that will convert from Fahrenheit back to Celsius.

Step 2: Replace Variables for Inverses

To find an inverse, swap the function notation: let \( y = f(x) = \frac{9}{5}x + 32 \). Our goal is to express \( x \) in terms of \( y \).

Step 3: Isolate the Variable

Starting with \( y = \frac{9}{5}x + 32 \), subtract 32 from both sides: \( y - 32 = \frac{9}{5}x \).

Step 4: Solve for the Original Input

To solve for \( x \), multiply both sides by \( \frac{5}{9} \): \( x = \frac{5}{9}(y - 32) \). This is the inverse function, which can be denoted as \( f^{-1}(y) = \frac{5}{9}(y - 32) \).

Step 5: Interpret the Inverse Function

The inverse function \( f^{-1}(y) = \frac{5}{9}(y - 32) \) takes a temperature in degrees Fahrenheit and converts it back to degrees Celsius.

Key Concepts

Celsius to Fahrenheit conversion

Temperature conversion between Celsius and Fahrenheit is a common task, especially in science classes. The function provided, \( f(x) = \frac{9}{5}x + 32 \), is used to convert a temperature from degrees Celsius to degrees Fahrenheit.

Each part of this function plays an important role:

• The fraction \( \frac{9}{5} \) is the scaling factor that adjusts the difference in size between the Celsius and Fahrenheit temperature units.
• The 32 added to the equation accounts for the difference in starting points of the two scales. While Celsius considers 0 as freezing point, Fahrenheit starts at 32 for freezing point.

This formula is very useful when you have to convert a given temperature in Celsius into its equivalent value in Fahrenheit. Knowing this formula is beneficial not only for academics but also for understanding weather reports from different parts of the world.

inverse operation

Inverse operations are essential in mathematics as they allow us to solve equations and understand functions from multiple perspectives. When we refer to inverse functions, we are essentially reversing the effect of the original function.

To find the inverse function, the roles of the input and output are swapped. If the original function converts Celsius degrees to Fahrenheit, the inverse function will convert Fahrenheit back to Celsius.

In our example, once we start with \( y = \frac{9}{5}x + 32 \) and solve for \( x \), we ultimately find the inverse function \( f^{-1}(y) = \frac{5}{9}(y - 32) \). This means that while the initial conversion formula tells you how to go from Celsius to Fahrenheit, the inverse formula is used to make the journey back to Celsius.

temperature conversion formula

Temperature conversion formulas allow for changes between different temperature measurement systems. The two most commonly used systems worldwide are Celsius and Fahrenheit. Each system has its unique reference points and scaling methods.

The Celsius to Fahrenheit conversion formula \( f(x) = \frac{9}{5}x + 32 \) takes into account these differences and gives a mathematical approach to changing one system to another.

When we talk about temperature conversion formulas, understanding the inverse formula is equally important. The inverse formula found here, \( f^{-1}(y) = \frac{5}{9}(y - 32) \), allows us to do the opposite by converting from Fahrenheit back to Celsius. These formulas ensure that regardless of which system is used, temperatures can be understood and compared equivalently. Knowing and understanding them makes it easier to adapt to the temperature system being used in various parts in the world.